<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>\(L^{p}\)-Approximation and generalized growth of generalized biaxially symmetric potentials on hyper sphere: \(L^{p}\)-Approximation and generalized growth of generalized biaxially symmetric potentials on hyper sphere</title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">


<div class="titlepage">
<h1>\(L^{p}\)-Approximation and generalized growth of generalized biaxially symmetric potentials on hyper sphere</h1>
<p class="authors">
<span class="author">Devendra Kumar\(^\ast \)</span>
</p>
<p class="date">January 30, 2017. Accepted: September 11, 2017. Published online: August 6, 2018.</p>
</div>
<p>\(^\ast \)Department of Mathematics, Research and Post Graduate Studies, M.M.H. College, Model Town, Ghaziabad-201001, U.P., India. Current Address: Department of Mathematics, Faculty of Sciences, Al-Baha University, P.O. Box 1988, Alaqiq, Al-Baha-65431, Saudi Arabia, K.S.A., e-mail: <span class="tt">d_kumar001@rediffmail.com</span> </p>

<div class="abstract"><p> The generalized order of growth and generalized type of an entire function \(F^{\alpha ,\beta }\) (generalized biaxisymmetric potentials) have been obtained in terms of the sequence \(E^p_n (F^{\alpha ,\beta },\Sigma ^{\alpha ,\beta }_r)\) of best real biaxially symmetric harmonic polynomial approximation on open hyper sphere \(\Sigma ^{\alpha ,\beta }_r\). Moreover, the results of McCoy [8] have been extended for the cases of fast growth as well as slow growth. </p>
<p><b class="bf">MSC.</b> 30B10, 41A10. </p>
<p><b class="bf">Keywords.</b> Generalized order and type, hyper sphere, generalized biaxisymmetric polynomial approximation errors, fast and slow growth, Jacobi polynomial and \(L^p\)-norm. </p>
</div>
<h1 id="a0000000002">1 INTRODUCTION</h1>
<p>Let \(F^{\alpha ,\beta }\) be a real valued regular solution to the generalized biaxisymmetric potential equation </p>
<div class="displaymath" id="a0000000003">
  \begin{eqnarray} \left(\tfrac {\partial ^2} {\partial x^2} + \tfrac {\partial ^2} {\partial y^2} + \tfrac {2\alpha +1} {x} \tfrac {\partial } {\partial x} + \tfrac {2 \beta +1} {y} \tfrac {\partial } {\partial y}\right) F^{\alpha ,\beta } = 0, \quad \alpha {\gt}\beta {\gt}-\tfrac {1}{2},\nonumber \end{eqnarray}
</div>
<p>where \((\alpha ,\beta )\) are fixed in a neighbourhood of the origin and the analytic Cauchy data \(F_x^{\alpha ,\beta }(0,y) = F_y^{\alpha +\beta } (x,0) = 0\) is satisfied along the singular lines in the open hyper sphere \(\Sigma ^{\alpha ,\beta }_r : x^2+y^2 {\lt} r^2\). Such functions with even harmonic functions are referred to as generalized biaxisymmetric potentials \((GBASP's)\) having local expansions of the form </p>
<div class="displaymath" id="a0000000004">
  \begin{eqnarray*} F^{\alpha ,\beta } (x,y) = \sum ^\infty _{n=0} a_n R_n^{\alpha ,\beta } (x,y) \end{eqnarray*}
</div>
<p>such that </p>
<div class="displaymath" id="a0000000005">
  \begin{eqnarray*} R_n^{\alpha ,\beta } (x,y) = (x^2 +y^2)^n P_n^{\alpha ,\beta } (x^2-y^2 / x^2+y^2) / P_n^{\alpha ,\beta } (1) , \quad n=0,1,2,\dots \end{eqnarray*}
</div>
<p>where the \(P_n^{\alpha ,\beta }\) are Jacobi polynomials <span class="cite">
	[
	<a href="#1" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#18" >18</a>
	]
</span>. </p>
<p>Let \(K\) be a compact subset of the complex plane. Let the one to one operator mapping between the space \(L^p (\Sigma ^{\alpha ,\beta }_r)\) of real valued GBASP’s with finite \(p\)-norm </p>
<div class="displaymath" id="a0000000006">
  \begin{eqnarray*} \| \cdot \| _p & =&  \left(\tfrac {1}{A} \iint _{ \Sigma ^{\alpha ,\beta }_r} |\cdot |^p dxdy\right)^{\tfrac {1}{p}}, \quad p\in [1,\infty ),\\ \| \cdot \| _\infty & =&  \sup _{ \Sigma ^{\alpha ,\beta }_r} |\cdot |, \quad \| 1\| _p = 1 \end{eqnarray*}
</div>
<p>and the space \(l^p(K)\) of associated functions </p>
<div class="displaymath" id="a0000000007">
  \begin{eqnarray*} f(z) = \sum ^\infty _{n=0} a_n z^{2n}, \quad R^{\alpha ,\beta }_n (z,0) = z^{2n}, \qquad n=0,1,2,\dots \end{eqnarray*}
</div>
<p>continuous on \(K\) with finite \(p\)-norm. Following McCoy <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> for Koornwinder’s integral for Jacobi polynomials and the inverse operator have been defined as: </p>
<div class="displaymath" id="a0000000008">
  \begin{eqnarray*} F^{\alpha ,\beta } (x,y) & =&  K_{\alpha ,\beta }(f) = \int ^1_0 \int ^\pi _0 f(\zeta ) \mu _{\alpha ,\beta } (t,s) dsdt\\ \mu _{\alpha ,\beta } (t,s) & =&  \gamma _{\alpha ,\beta } (1-t^2)^{\alpha -\beta -1} t^{2\beta +1} (\sin \  s)^{2\alpha }\\ \zeta ^2 & =&  x^2-y^2 t^2-i2xyt\cos \  s\\ \gamma _{\alpha ,\beta } & =&  2\Gamma (\alpha +1) / \Gamma (\tfrac {1}{2}) \Gamma (\alpha -\beta ) \Gamma (\beta +\tfrac {1}{2}), \end{eqnarray*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000009">
  \begin{eqnarray*} f(z) & =&  K^{-1}_{\alpha ,\beta } (F^{\alpha ,\beta }) = \int ^1_{-1} F^{\alpha ,\beta } (r\xi ,r(1-\xi ^2)^{\tfrac {1}{2}}) \nu _{\alpha ,\beta } ((z/r^{2})^2,\xi ) d\xi \\ \nu _{\alpha ,\beta }(\tau ,\xi ) & = &  S_{\alpha ,\beta } (\tau ,\xi ) (1-\xi )^\alpha (1+\xi )^\beta \\ S_{\alpha ,\beta } (\tau ,\xi ) & =&  \eta _{\alpha ,\beta } \tfrac {1-\tau } {(1+\tau )^{ \alpha +\beta +2}} F\left(\tfrac {\alpha +\beta +2}{2}; \tfrac {\alpha +\beta +3}{2}; \beta +1; \tfrac {2\tau (1+\xi )} {(1+\tau )^2}\right)\\ \eta _{\alpha ,\beta } & =&  \Gamma (\alpha +\beta +2) / 2^{\alpha +\beta +1} \Gamma (\alpha +1) \Gamma (\beta +1). \end{eqnarray*}
</div>
<p>The normalizations \(K_{\alpha ,\beta } (1) = K^{-1}_{\alpha ,\beta }(1)=1\) are taken. The kernel \(S_{\alpha ,\beta } (\tau ,\xi )\) is analytic on \(\| \tau \| {\lt}1\) for \(-1\le \xi \le 1\). The local function elements \(F^{\alpha ,\beta }\) and \(f\) are continued harmonically/analytically by contour deformation using the Envelope Method <span class="cite">
	[
	<a href="#3" >3</a>
	]
</span>. </p>
<p>Let \(K\) be a compact subset of the complex plane with Card. \(K = 0\) and let \(u_1,u_2,\dots ,u_n \in K\). Following <span class="cite">
	[
	<a href="#4" >4</a>
	, 
	p.285
	]
</span> we put </p>
<div class="displaymath" id="a0000000010">
  \begin{eqnarray*} & &  V(u_1,u_2,\dots ,u_n) = \textstyle \prod \limits ^n_{k,l(k{\lt}l)} (u_k-u_l), \\ & &  V_n = \max \{ |V(u_1,u_2,\dots ,u_n)|:u_k \in K, 1\le k\le n\} . \end{eqnarray*}
</div>
<p>Set \(d=\max \{ |z|:z\in K\} \). Also, let \(\mu _n(z) = z^n + a_1z^{n-1} + \dots +a_n\) denote the Chebyshev polynomial for \(K\) such that all zeros of \(\mu _n\) belong to \(K\). We set </p>
<div class="displaymath" id="a0000000011">
  \begin{eqnarray*} m^*_n = \max \{ |\mu _n(z)|, z\in K\} . \end{eqnarray*}
</div>
<p>Then we have <span class="cite">
	[
	<a href="#4" >4</a>
	, 
	pp.
	
	287–289
	]
</span>, </p>
<div class="displaymath" id="a0000000012">
  \begin{eqnarray*} & &  m^*_n \le \tfrac {V_{n+1}}{V_n} \le (n+1) m^*_n,\\ & &  \lim _{n\rightarrow \infty } \left(\tfrac {V_{n+1}}{V_n}\right)^{1/n} = \widetilde{d}, \end{eqnarray*}
</div>
<p>where \(\widetilde{d}\) is the transfinite diameter of \(K\). </p>
<p>Using the Koornwinder’s integral and inverse operator the information concerning the approximation and growth of analytic functions can be transfer to GBASP \(F^{\alpha ,\beta }\). </p>
<p>The essential properties of \(F^{\alpha ,\beta } \in L^p (\Sigma ^{\alpha ,\beta }_r)\) that are the restrictions of entire GBASP functions are drawn from approximation on sets of polynomials </p>
<div class="displaymath" id="a0000000013">
  \begin{eqnarray*} P^{\alpha ,\beta }_{2n} = \{ K_{\alpha ,\beta } (h):h\in p_{2n}\} ,\end{eqnarray*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000014">
  \begin{eqnarray*} p_{2n} = \left\{ \sum ^n_{k=0}a_k z^{2k} : a_k \textrm{ - real}, 0\le k\le n\right\} , \quad n=0,1,2,\dots . \end{eqnarray*}
</div>
<p>It is the Bernstein limits of the optimal approximates, </p>
<div class="displaymath" id="a0000000015">
  \begin{eqnarray*} E^p_{2n} = E^{p}_{2n} (F^{\alpha ,\beta }, \textstyle \Sigma ^{\alpha ,\beta }_r) = \min \left\{  \|  F^{\alpha ,\beta } = H\| _p : H\in P^{\alpha ,\beta }_{2n}\right\} , \end{eqnarray*}
</div>
<p>and </p>
<div class="displaymath" id="a0000000016">
  \begin{eqnarray*} e^p_{2n} = e^p_{2n} (f;k) = \min \left\{ \| f-h\| _p :h\in p_{2n}\right\} , \end{eqnarray*}
</div>
<p>and provide the characterizations. The set \(p_{2n}\) contains all real polynomials of degree at most \(2n\) and set \(P^{\alpha ,\beta }_{2n}\) contains all real biaxisymmetric harmonic polynomials of degree at most \(2n\). The operators \(K_{\alpha ,\beta }\) and \(K^{-1}_{\alpha ,\beta }\) establish one-one equivalence of sets \(p_{2n}\) and \(P^{\alpha ,\beta }_{2n}\). </p>
<p>Several authors such as Harfaoui <span class="cite">
	[
	<a href="#5" >5</a>
	]
</span>, Kumar <span class="cite">
	[
	<a href="#10" >10</a>
	]
</span>, Harfaoui and Kumar <span class="cite">
	[
	<a href="#6" >6</a>
	]
</span> and others obtained generalized characteristics of growth of entire functions by using the best polynomial approximation and interpolation in \(L_{p}\)-norm. The growth characteristics of solutions of certain linear partial differential equations have been studied by Kumar and Basu <span class="cite">
	[
	<a href="#12" >12</a>
	]
</span>, <span class="cite">
	[
	<a href="#13" >13</a>
	]
</span>, Kumar <span class="cite">
	[
	<a href="#11" >11</a>
	]
</span>, Khan and Ali <span class="cite">
	[
	<a href="#9" >9</a>
	]
</span>. </p>
<p>McCoy <span class="cite">
	[
	<a href="#15" >15</a>
	, 
	Th.
	
	2
	]
</span> obtained the necessary and sufficient conditions for the entire GBASP \(F^{\alpha ,\beta } \in L^p(D), p\ge 2\) to be the restriction to \(D\) of order and type in terms of the errors \(E^p_n(F^{\alpha ,\beta })\), here \(D\) is parabolic convex set. To the best of our knowledge, these characterizations leave an important class of growth of entire function GBASP \(F^{\alpha ,\beta }\) such as fast and slow growth. In this paper we have tried to fill this gap. Moreover, we have extended the results of McCoy to generalized orders and generalized types which will cover the cases of fast growth as well as slow growth. Here we replace \(D\) by open hypersphere \(\Sigma ^{\alpha ,\beta }_r\). </p>
<h1 id="a0000000017">2 GENERALIZED ORDER AND GENERALIZED TYPE</h1>
<p> Seremeta <span class="cite">
	[
	<a href="#17" >17</a>
	]
</span> defined the generalized order and generalized type with the help of general functions as follows. </p>
<p>Let \(L^0\) denote the class of functions \(h\) satisfying the following conditions: </p>
<ol class="enumerate">
  <li><p>\(\phi (x)\) is defined on \([a,\infty )\) and is positive, strictly increasing, differentiable and tends to \(\infty \) as \(x\rightarrow \infty \). </p>
</li>
  <li><p>It holds </p>
<div class="displaymath" id="a0000000018">
  \begin{eqnarray*} \lim _{x\rightarrow \infty } \tfrac {\phi \left\{ \left(1+\frac{1}{\varphi (x)}\right) (x)\right\} } {\phi (x)} = 1 \end{eqnarray*}
</div>
<p>for every function \(\varphi (x)\) such that \(\varphi (x)\rightarrow \infty \), as \(x\rightarrow \infty \). </p>
<p>Let \(\Delta \) denote the class of functions \(\phi \) satisfying condition (i) and the following: </p>
</li>
  <li><p>It holds </p>
<div class="displaymath" id="a0000000019">
  \begin{eqnarray*} \lim _{x\rightarrow \infty } \tfrac {\phi (c x)} {\phi (x)} = 1 \end{eqnarray*}
</div>
<p>for every \(c{\gt}0\), that is, \(\phi (x)\) is slowly increasing. </p>
<p>For an entire function \(f(z)\) and functions \(\alpha (x)\in \Delta ,\beta (x)\in L^0\), Seremeta <span class="cite">
	[
	<a href="#10" >10</a>
	, 
	Th.
	
	1
	]
</span> proved that </p>
<div class="displaymath" id="a0000000020">
  \begin{eqnarray*} \rho (\alpha ,\beta ,f) = \limsup _{r\rightarrow \infty } \tfrac {\alpha \left[\log M(r,f)\right]} {\beta (\log r)} = \limsup _{n\rightarrow \infty } \tfrac {\alpha (n)} {\beta \big(-\tfrac {1}{n} \log |a_n|\big)}. \end{eqnarray*}
</div>
<p>Further, for \(\alpha (x)\in L^0, \beta ^{-1}(x) \in L^0\) and \(\gamma (x) \in L^0\), we have </p>
<div class="displaymath" id="a0000000021">
  \begin{eqnarray*} T(\alpha ,\beta ,f) = \limsup _{r\rightarrow \infty } \tfrac {\alpha \left[\log M(r,f)\right]} {\beta \left[\left(\gamma (r)\right)^\rho \right]} = \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{n}{\rho })} {\beta \Big[\gamma (e^{\frac{1}{\rho }} |a_n|^{-\frac{1}{n}})\Big]^\rho }, \end{eqnarray*}
</div>
<p>where \(0{\lt}\rho {\lt}\infty \) is a fixed number. </p>
<p>The above relations were obtained under certain conditions which do not hold if \(\alpha = \beta \). In 1968, Seremeta <span class="cite">
	[
	<a href="#16" >16</a>
	]
</span> obtained the results connected with slow growth of entire functions. The characteristic for slow growth entire functions \(f(z) = \sum ^\infty _{k=0}c_k(f) z^k\) has the following form: </p>
<div class="displaymath" id="a0000000022">
  \begin{eqnarray*} \rho _\alpha = \limsup _{n\rightarrow \infty } \frac{\alpha \left(\log \log M(r,f)\right)}{\alpha (\log \log r)}, \end{eqnarray*}
</div>
<p>where \(M(r,f) = \max _{|z|=r} |f(z)|\) and a function \(\alpha \in \Delta \). Let us define \(F(x,c) = \alpha ^{-1}(c\alpha (x))\), where \(c\in (0,\infty )\) is any constant. It was shown in [16] that if for any \(c\in (0,\infty )\) the inequality </p>
<div class="displaymath" id="a0000000023">
  \begin{eqnarray*} 0\le \tfrac {dF(x,c)}{dx} \le A\big(\exp (F(x,c))^B\big) \end{eqnarray*}
</div>
<p>is realized for any \(x\ge x_1\), where \(A\) and \(B\) are some constants \((0{\lt}A,B{\lt}\infty )\), then we obtain </p>
<div class="displaymath" id="a0000000024">
  \begin{eqnarray*} \rho _\alpha = \max (\rho ^\prime _\alpha ,1).\end{eqnarray*}
</div>
<p>Here </p>
<div class="displaymath" id="a0000000025">
  \begin{eqnarray*} \rho ^\prime _\alpha = \limsup _{k\rightarrow \infty } \tfrac {\alpha (\log k)} {\alpha \left(\log \left(\frac{1}{k} \log |c_k(f)|^{-1}\right)\right)}. \end{eqnarray*}
</div>
<p>To refine this scale, Kapoor and Nautiyal <span class="cite">
	[
	<a href="#7" >7</a>
	]
</span> introduced a new class of functions as follows: A function \(\phi (t)\in \Omega \) if \(\phi (t)\) satisfies (ii) and: </p>
</li>
  <li><p>There exists a function \(\delta (t) \in \Delta \) and \(t_0,K_1\) and \(K_2\) such that for all \(t{\gt}t_0\) </p>
<div class="displaymath" id="a0000000026">
  \begin{eqnarray*} 0{\lt}K_1 \le \tfrac {d(\phi (t))} {d(\delta (\log t))} \le K_2 {\lt}\infty . \end{eqnarray*}
</div>
<p>Further a function \(\phi (t)\in \overline{\Omega }\) if \(\phi (t)\) satisfies (ii) and </p>
</li>
  <li><div class="displaymath" id="a0000000027">
  \begin{eqnarray*} \lim _{t\rightarrow \infty } \tfrac {d(\phi (t))} {d(\log (t))} = K, \quad 0{\lt}K{\lt}\infty .\end{eqnarray*}
</div>
<p>Kapoor and Nautiyal <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	p.
	
	66
	]
</span> showed that \(\Omega ,\overline{\Omega }\subseteq \Delta \) and \(\Omega \cap \overline{\Omega }=\Phi \). Let \(\alpha (t)\in \Omega \) or \(\overline{\Omega }\). Then following Kapoor and Nautiyal <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	p.
	
	66
	]
</span>, for entire GBASP \(F^{\alpha ,\beta }\) and associate we define the generalized order and generalized type as </p>
<div class="equation" id="b1">
<p>
  <div class="equation_content">
    \begin{equation} \rho = \rho (\alpha ,\alpha ,f) = \limsup _{r\rightarrow \infty } \tfrac {\alpha (\log M(R,f))} {\alpha (\log r)} \label{b1}\end{equation}
  </div>
  <span class="equation_label">2.1</span>
</p>
</div>
<div class="equation" id="b2">
<p>
  <div class="equation_content">
    \begin{equation} T = T(\alpha ,\alpha ,f) = \limsup _{r\rightarrow \infty } \tfrac {\alpha [ \log M(r,f)]} {[\alpha (\log r)^\rho ]} \label{b2} \end{equation}
  </div>
  <span class="equation_label">2.2</span>
</p>
</div>
<div class="equation" id="b3">
<p>
  <div class="equation_content">
    \begin{equation} \rho ^* = \rho ^* (\alpha ,\alpha , F^{\alpha ,\beta }) = \limsup _{r\rightarrow \infty } \tfrac {\alpha (\log M(r,F^{\alpha ,\beta }))} {\alpha (\log r)} \label{b3} \end{equation}
  </div>
  <span class="equation_label">2.3</span>
</p>
</div>
<div class="equation" id="b4">
<p>
  <div class="equation_content">
    \begin{equation} T^* = T^* (\alpha ,\alpha ,F^{\alpha ,\beta }) = \limsup _{r\rightarrow \infty } \tfrac {\alpha [\log M(r,F^{\alpha ,\beta })]} {[\alpha (\log r)^\rho ]}, \label{b4}\end{equation}
  </div>
  <span class="equation_label">2.4</span>
</p>
</div>
<p>where </p>
<div class="displaymath" id="a0000000028">
  \begin{eqnarray*} M(r,f) = \max _{|z|=r} |f(z)|, M(r,F^{\alpha ,\beta }) = \max _{x^2+y^2=r^2} |F^{\alpha ,\beta } (x,y)|. \end{eqnarray*}
</div>
<p>Let \(K_r\) be the largest equipotential curve of \(K\) defined by \(K_r = \{ z\in \mathbb {C}:|\gamma (z)| d = r\} \), where \(w=\gamma (z)\) is holomorphic and maps the unbounded component of the complement of \(K\) on \(|w|{\gt}1\) such that \(\gamma (\infty ) = \infty \) and \(\gamma ^\prime (\infty ){\gt}0\). When \(r=d=1,K_r=K\). So we take \(r{\gt}d,r{\gt}1\). We set \(\overline{M} (r,F^{\alpha ,\beta }) = \sup _{z\in K_r} |F^{\alpha ,\beta } (z,o)|\) for \(r{\gt}1\). </p>
</li>
</ol>
<p> McCoy <span class="cite">
	[
	<a href="#14" >14</a>
	]
</span> proved the following result: <div class="t1_thmwrapper " id="a0000000029">
  <div class="t1_thmheading">
    <span class="t1_thmcaption">
    Theorem
    </span>
    <span class="t1_thmlabel">2.1</span>
  </div>
  <div class="t1_thmcontent">
  <p>For each GBASP \(F^{\alpha ,\beta }\) regular in the hyper sphere \(\Sigma ^{\alpha ,\beta }_r\) there is a unique \(K_{\alpha ,\beta }\) associated even function \(f\) analytic in the disk \(D_r\) and conversely. </p>

  </div>
</div> Now we prove the following Lemmas: </p>
<p><div class="l1_thmwrapper " id="a0000000030">
  <div class="l1_thmheading">
    <span class="l1_thmcaption">
    Lemma
    </span>
    <span class="l1_thmlabel">2.1</span>
  </div>
  <div class="l1_thmcontent">
  <p>Let \(F^{\alpha ,\beta }\) be real valued entire function GBASP with \(K_{\alpha ,\beta }\) associate \(f\). Then the generalized order and generalized type of \(F^{\alpha ,\beta }\) respectively are identical. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000031">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let us consider the relation \(F^{\alpha ,\beta }(x,y) = K_{\alpha ,\beta }(f)\). The nonnegativity and the normalization of the measure leads directly to the bound </p>
<div class="equation" id="b5">
<p>
  <div class="equation_content">
    \begin{equation} M(r,F^{\alpha ,\beta }) \le M(r,f). \label{b5} \end{equation}
  </div>
  <span class="equation_label">2.5</span>
</p>
</div>
<p>The inverse relation </p>
<div class="displaymath" id="a0000000032">
  \begin{eqnarray*} f(z) = K^{-1}_{\alpha ,\beta } (F^{\alpha ,\beta }) \end{eqnarray*}
</div>
<p>leads to the inequality </p>
<div class="displaymath" id="a0000000033">
  \begin{eqnarray*} |f(z)| \le M(r,F^{\alpha ,\beta }) N_{\alpha ,\beta } (\tau ), \quad \tau = \left(\tfrac {z}{r}\right)^2, \end{eqnarray*}
</div>
<p>where </p>
<div class="displaymath" id="a0000000034">
  \begin{eqnarray*} N_{\alpha ,\beta }(\tau ) = \max \{ \eta ^{-1}_{\alpha ,\beta } | S_{\alpha ,\beta } (\tau ,\xi )| ; -1 \le \xi \le 1\} . \end{eqnarray*}
</div>
<p>However, for \(z=\varepsilon r e^{i\theta } (\varepsilon \) real), </p>
<div class="displaymath" id="a0000000035">
  \begin{eqnarray*} M(\varepsilon r,f) \le M(r,F^{\alpha ,\beta }) N_{\alpha ,\beta }(\tau ) \end{eqnarray*}
</div>
<p>it gives </p>
<div class="equation" id="b6">
<p>
  <div class="equation_content">
    \begin{equation} M(r,f) \le M(\varepsilon ^{-1}r, F^{\alpha ,\beta }) N_{\alpha ,\beta }(\varepsilon ^2). \label{b6} \end{equation}
  </div>
  <span class="equation_label">2.6</span>
</p>
</div>
<p>Using inequalities (<a href="#b5">2.5</a>), (<a href="#b6">2.6</a>) and definitions (2.1)–(2.4), the proof is completed. </p>
<p><div class="l1_thmwrapper " id="a0000000036">
  <div class="l1_thmheading">
    <span class="l1_thmcaption">
    Lemma
    </span>
    <span class="l1_thmlabel">2.2</span>
  </div>
  <div class="l1_thmcontent">
  <p>Let \(F^{\alpha ,\beta }\) be a real entire function GBASP of generalized order and generalized type. Then </p>
<div class="equation" id="b7">
<p>
  <div class="equation_content">
    \begin{equation} \rho = \limsup _{r\rightarrow \infty } \tfrac {\alpha (\log \overline{M}(r,F^{\alpha ,\beta }))} {\alpha (\log r)} \label{b7} \end{equation}
  </div>
  <span class="equation_label">2.7</span>
</p>
</div>
<div class="equation" id="b8">
<p>
  <div class="equation_content">
    \begin{equation} T = \limsup _{r\rightarrow \infty } \tfrac {\alpha [\log \overline{M} (r,F^{\alpha ,\beta })]} {[\alpha (\log r)^\rho ].} \label{b8} \end{equation}
  </div>
  <span class="equation_label">2.8</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000037">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Using the definitions of generalized order and generalized type of entire GBASP and proof proceeds on the lines of Lemma 1 <span class="cite">
	[
	<a href="#6" >6</a>
	]
</span>. </p>
<p><div class="l1_thmwrapper " id="a0000000038">
  <div class="l1_thmheading">
    <span class="l1_thmcaption">
    Lemma
    </span>
    <span class="l1_thmlabel">2.3</span>
  </div>
  <div class="l1_thmcontent">
  <p>Let \(\alpha (x)\in \overline{\Omega }\) and \(K\subseteq \mathbb {C}\) be an arbitrary compact set with \(\operatorname {card} K=\infty \). Let \(f\in L^p(K), 1\le p \le \infty \), be an entire function. Then \(f\) has generalized order \(\rho (f), 1\le \rho (f)\le \infty \), if and only if, </p>
<div class="equation" id="b9">
<p>
  <div class="equation_content">
    \begin{equation} \rho (f) = \Theta (L(f)), \label{b9} \end{equation}
  </div>
  <span class="equation_label">2.9</span>
</p>
</div>
<p>where </p>
<div class="equation" id="b10">
<p>
  <div class="equation_content">
    \begin{equation} L(f) = \limsup _{n\rightarrow \infty } \tfrac {\alpha (n)} {\alpha [\log \{ e^p_n(f,K)/m^*_{n+1}\} ^{-\frac{1}{n}}]}, \label{b10} \end{equation}
  </div>
  <span class="equation_label">2.10</span>
</p>
</div>
<p>and </p>
<div class="displaymath" id="a0000000039">
  \begin{eqnarray*} \Theta (L(f)) = \begin{cases}  \max \{ 1,L(f)\} , &  \textrm{if \  } \alpha (x)\in \Omega , \\ 1+L(f), &  \textrm{if \  } \alpha (x)\in \overline{\Omega }. \end{cases}\end{eqnarray*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000040">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(d=\max \{ |z|;z\in K\} \). From the definition of \(e^p_n(f,K)\), since \(h\in p_n\), we have </p>
<div class="equation" id="b11">
<p>
  <div class="equation_content">
    \begin{equation} e^p_n (f,K) \le \| f-h\| _p \le A^{\frac{1}{p}} \max _{z\in K} |f(z)-h(z)| = A^{\frac{1}{p}} e_n(f,K), \label{b11} \end{equation}
  </div>
  <span class="equation_label">2.13</span>
</p>
</div>
<p>where \(A\) is the area of \(K\). </p>
<p>Now using Lemma 2 <span class="cite">
	[
	<a href="#2" >2</a>
	, 
	p.
	
	923
	]
</span>, it has been shown that for \(R{\gt}d\), </p>
<div class="equation" id="b12">
<p>
  <div class="equation_content">
    \begin{equation} e_n(f,K) \le \tfrac {Rm^*_{n+1}} {(R-d)^{n+2}} \tfrac {1}{2\pi } \int ^{2\pi }_0 |f(Re^{i\theta })|d\theta . \label{b12} \end{equation}
  </div>
  <span class="equation_label">2.14</span>
</p>
</div>
<p>Using (<a href="#b12">2.14</a>) in (<a href="#b11">2.13</a>) we get </p>
<div class="equation" id="b13">
<p>
  <div class="equation_content">
    \begin{equation} e^p_n(f,K)/m^*_{n+1} \le A^{\frac{1}{p}} R(R-d)^{-(n+2)} M(R,f). \label{b13} \end{equation}
  </div>
  <span class="equation_label">2.15</span>
</p>
</div>
<p>Suppose \(\alpha (x)\in \overline{\Omega }\) and \(\rho {\lt}\infty \). Then by the definition of \(\rho \), we have for any given \(\varepsilon {\gt}0\) and \(R{\gt}R_0(\varepsilon ), R(\varepsilon )\in (0,\infty )\), </p>
<div class="equation" id="b14">
<p>
  <div class="equation_content">
    \begin{equation} \alpha (\log M(R,f)) \le \alpha (\log R) \overline{\rho },\quad \overline{\rho } = \rho +\varepsilon . \label{b14} \end{equation}
  </div>
  <span class="equation_label">2.16</span>
</p>
</div>
<p>In view of (<a href="#b13">2.15</a>) and (<a href="#b14">2.16</a>) we have </p>
<div class="equation" id="b15">
<p>
  <div class="equation_content">
    \begin{equation} e^p_n(f,K)/m^*_{n+1} < A^{\frac{1}{p}} R(R-d)^{-(n+2)} \exp [\alpha ^{-1}\{ \overline{\rho }\alpha (\log R)\} ]. \label{b15} \end{equation}
  </div>
  <span class="equation_label">2.17</span>
</p>
</div>
<p>Since \(d\) is finite and fixed and the above inequality holds for all \(R{\gt}R_0 (\varepsilon )\), we can choose </p>
<div class="displaymath" id="a0000000041">
  \begin{eqnarray*} R=R(n) = \exp \left[\alpha ^{-1} \left\{  \tfrac {\alpha (n)} {\overline{\rho }-1}\right\} \right] = \exp \left[ F\big(n,\tfrac {1}{\rho -1}\big)\right]. \end{eqnarray*}
</div>
<p>Substituting this value of \(R\) in (<a href="#b15">2.17</a>), we obtain </p>
<div class="displaymath" id="a0000000042">
  \begin{eqnarray*} e^p_n (f,K)/m^*_{n+1} & {\lt}&  A^{\frac{1}{p}} \exp \left[-(n+1) F\big(n,\tfrac {1}{\rho -1}\big)\right] \exp \left[\alpha ^{-1} \left\{ \overline{\rho }+ \tfrac {\alpha (n)} {\overline{\rho }-1}\right\} \right]\\ & {\lt}&  A^{\frac{1}{p}} \exp \left[-n\left\{ F\big(n,\tfrac {1}{\overline{\rho }-1}\big)-1\right\} \right], \end{eqnarray*}
</div>
<p>since \(F\big(n,\tfrac {1}{\overline{\rho }-1}\big)\rightarrow \infty \) as \(n\rightarrow \infty \). Hence </p>
<div class="displaymath" id="a0000000043">
  \begin{align*} &  \log \left[\left\{ e^p_n(f,K)/m^*_{n+1}\right\} ^{-\frac{1}{n}}\right]{\gt}\\ & {\gt} F\big(n,\tfrac {1}{\overline{\rho }-1}\big)-1 -\tfrac {1}{np} \log A\\ & = \alpha ^{-1} \left\{ \tfrac {\alpha (n)} {\overline{\rho }-1}\right\}  \left\{ 1-\left(F\big(n,\tfrac {1}{\overline{\rho }-1}\big)\right)^{-1} \left[1+\tfrac {1}{np}\log A\right]\right\}  {\gt} \tfrac {\alpha (n)} {\overline{\rho }-1}. \end{align*}
</div>
<p>Since \(\alpha (x)\in \overline{\Omega }\), as \(n\rightarrow \infty \), we have </p>
<div class="equation" id="b16">
<p>
  <div class="equation_content">
    \begin{equation} \rho (f) \ge 1+\limsup _{n\rightarrow \infty } \tfrac {\alpha (n)} {\alpha \big[\log [e^p_n (K,f)/m^*_{n+1}]^{-\frac{1}{n}}\big]}. \label{b16} \end{equation}
  </div>
  <span class="equation_label">2.18</span>
</p>
</div>
<p>In order to prove reverse inequality, let us put </p>
<div class="equation" id="b17">
<p>
  <div class="equation_content">
    \begin{equation} \limsup _{n\rightarrow \infty } \tfrac {\alpha (n)} {\alpha \big[\log \{ e^p_n (f,K)/m^*_{n+1}\} ^{-\frac{1}{n}}\big]} = L(f). \label{b17} \end{equation}
  </div>
  <span class="equation_label">2.19</span>
</p>
</div>
<p>Suppose \(L(f){\lt}\infty \). Then for given \(\varepsilon {\gt} 0\) and all \(n{\gt}n_0(\varepsilon )\), we have </p>
<div class="displaymath" id="a0000000044">
  \begin{eqnarray*} e^p_n(f,K){\lt}m^*_{n+1} \exp \big[-nF\big(n,\tfrac {1}{\overline{L}(f)}\big)\big], \quad \overline{L}(f) = L(f)+\varepsilon . \end{eqnarray*}
</div>
<p>Now we consider the function </p>
<div class="displaymath" id="a0000000045">
  \begin{eqnarray*} h^*(z) = \sum ^\infty _{n=n_0} a^{n+1}_{n+1,R_0} z^{n+1} \exp \Big[-n F\Big(n,\tfrac {1}{\overline{L}(f)}\Big)\Big] \end{eqnarray*}
</div>
<p>where \(R_0{\gt}d\) and </p>
<div class="displaymath" id="a0000000046">
  \[  a_{n+1,R_0} = \left\{ \left(1+\tfrac {d}{R_0}\right)^{n+2} \left[ \tfrac {2(n+2)} {1-\tfrac {d}{R_0}}\right]\right\} ^{\frac{1}{(n+1)}}.  \]
</div>
<p> Suppose \(\{ p_n(z)\} ^\infty _0\) be the best polynomial approximating for the function \(f\) on \(K\). Let \(D_R\) denote the disk of radius \(R\) centered at the origin and \(\Gamma _{R}\) be the boundary of \(D_R\). Let </p>
<div class="equation" id="b18">
<p>
  <div class="equation_content">
    \begin{equation} S(z) = \sum ^\infty _{n=0}\left\{ p_{n+1}(z) - p_n(z)\right\}  + p_0(z). \label{b18} \end{equation}
  </div>
  <span class="equation_label">2.20</span>
</p>
</div>
<p>In view of proof of <span class="cite">
	[
	<a href="#2" >2</a>
	, 
	Th.
	
	1,
	
	p.
	
	924
	]
</span>, it can be easily seen that the series (<a href="#b18">2.20</a>) is uniformly convergent on \(\Gamma _{R}\) for any arbitrary \(R{\gt}0\). Thus the sum represent an entire function. Now </p>
<div class="displaymath" id="a0000000047">
  \begin{eqnarray*} S(z) & =&  \lim _{n\rightarrow \infty } \left\{  \sum ^n_{m=0} \left\{  p_{m+1} (z) - p_m (z)\right\} \right\}  + p_0 (z),\\ & =&  \lim _{n\rightarrow \infty } p_{n+1}(z) = f(z). \end{eqnarray*}
</div>
<p>On \(K\subseteq \mathbb {C}\) we have the inequality </p>
<div class="displaymath" id="a0000000048">
  \begin{eqnarray*} \| p_{n+1}(z) - p_n(z)\| _p & \le &  \| p_{n+1} (z) - f(z)\| _p + \| p_n(z) - f(z) \| _p\\ & \le &  2 e^p_n (f,K). \end{eqnarray*}
</div>
<p>Dovgoshei <span class="cite">
	[
	<a href="#2" >2</a>
	, 
	p.
	
	924
	]
</span> shown that </p>
<div class="displaymath" id="a0000000049">
  \begin{eqnarray*} \max _{z\in \Gamma _{R}} |p_{n+1}(z) - p_n(z) |^{\frac{1}{n}} & \le &  \left[[R^{n+1} e^\infty _n (f,K)/m^*_{n+1}]\right]^{\frac{1}{n}}\\ & & \cdot \left[ 2(n+2) \tfrac {(1+(\frac{d}{R}))^{n+2}} {1-\frac{d}{R}}\right]^{\frac{1}{n}}. \end{eqnarray*}
</div>
<p>For \(1\le p\le \infty \), we have </p>
<div class="displaymath" id="a0000000050">
  \begin{eqnarray*} \| p_{n+1}(z)-p_n(z)\| _p \le \left[2A R^{n+1} (n+2) \tfrac {\big(1+(\frac{d}{R})\big)^{n+2}}{m^*_{n+1} (1-\frac{d}{R})}\right]^{\frac{1}{n p}} [e^p_n (f,K)]^{\frac{1}{n}} \end{eqnarray*}
</div>
<p>leading to the relation </p>
<div class="displaymath" id="a0000000051">
  \begin{eqnarray*} \max _{z\in \Gamma _{R}} |h^*(z)| & =&  \sum ^\infty _{n=n^\prime } a^{n+1}_{n+1,R_0} R^{n+1} \exp \Big[-n F\big(n,\tfrac {1} {\overline{L}(f)}\big)\Big]\\ & \ge &  \max _{z\in \Gamma _{R}} |S(z)-p_{n^\prime }(z)| \ge A^{-\tfrac {1}{p}} \| S(z) - p_{n^\prime }(z)\| _p. \end{eqnarray*}
</div>
<p>From the last inequality, using <span class="cite">
	[
	<a href="#7" >7</a>
	, 
	Th.
	
	4
	]
</span>, the relationship between generalized order \(\rho \) for \(\alpha (x)\in \overline{\Omega }\) and the Taylor coefficients of the function \(h^*(z)\), we obtain generalized order of \(h^*(z)\ge \) generalized order of \(f\). </p>
<p>If \(\rho _{1}\) denotes the generalized order of \(h^*(z)\) then </p>
<div class="displaymath" id="a0000000052">
  \begin{eqnarray*} \rho _1 = 1+\limsup _{n\rightarrow \infty } \tfrac {\alpha (n+1)} {\alpha \Big[\log \Big\{ \big|a^{n+1}_{n+1,R_0} \big| \exp \big\{ -nF\big(n,\frac{1}{\overline{L}(f)}\big)\big\} \Big\} ^{-\frac{1}{(n+1)}}\Big]}. \end{eqnarray*}
</div>
<p>Since </p>
<div class="displaymath" id="a0000000053">
  \begin{eqnarray*} \log (a_{n+1,R_0}) = \tfrac {n+2}{n+1} \log (1+\tfrac {d}{R_0}) + \tfrac {1}{n+1} \log \left(\tfrac {2(n+1)}{1-\frac{d}{R_0}}\right) = \mathcal{O}(1), \  \textrm{ as } n\rightarrow \infty . \end{eqnarray*}
</div>
<p>Hence </p>
<div class="displaymath" id="a0000000054">
  \begin{align*} & \log \big\{ |a^{n+1}_{n+1,R_0}| \exp \big\{ -n F \big(n, \tfrac {1}{\overline{L}(F)}\big)\big\} \big\} ^{-\frac{1}{(n+1)}}=\\ & = -\tfrac {1}{n+1} \log \big[\exp \big(-nF\big(n,\tfrac {1}{\overline{L} (F)}\big)\big)\big]-\mathcal{O}(1)\\ & = \tfrac {n}{n+1} F \big(n,\tfrac {1}{\overline{L}(f)}\big) \simeq \alpha ^{-1} \Big[ \tfrac {\alpha (n)}{\overline{L}(f)} \Big] -\mathcal{O}(1). \end{align*}
</div>
<p>Since \(\alpha (x)\in \overline{\Omega }\), we finally get </p>
<div class="displaymath" id="a0000000055">
  \begin{eqnarray*} \rho _1 = 1+\limsup _{n\rightarrow \infty } \tfrac {\alpha (n+1) \overline{L}(f)} {\alpha (n)} = 1+\overline{L}(f) = 1+L(f)+\varepsilon . \end{eqnarray*}
</div>
<p>Since \(\varepsilon \) was arbitrary, we get \(\rho _1\ge \rho (f)\). Combining this with (<a href="#b16">2.18</a>) we get (<a href="#b9">2.9</a>). </p>
<p><div class="l1_thmwrapper " id="a0000000056">
  <div class="l1_thmheading">
    <span class="l1_thmcaption">
    Lemma
    </span>
    <span class="l1_thmlabel">2.4</span>
  </div>
  <div class="l1_thmcontent">
  <p>Let \(\alpha (x)\in \overline{\Omega }\) and \(K\subseteq \mathbb {C}\) be an arbitrary compact set with Card. \(K=\infty \). Let \(f\in L^p(K), 1\le p\le \infty \), be an entire function. Then \(f\) has generalized order \(\rho (f)\) and finite generalized type \(T(f)\) if, and only if, </p>
<div class="equation" id="b19">
<p>
  <div class="equation_content">
    \begin{equation} T=T(f) = \limsup _{n\rightarrow \infty } \tfrac {\alpha (\log M(R,f))} {[\alpha (\log R)]^\rho } = \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{n}{\rho })} {\Big\{ \alpha \big[\frac{\rho }{\rho -1} \log \big[\frac{e^p_n(f,K)}{m^*_{m+1}}\big]^{-\frac{1}{n}}\big]\Big\} ^{\rho +1}}\label{b19} \end{equation}
  </div>
  <span class="equation_label">2.21</span>
</p>
</div>
<p>provided \(\tfrac {d[\alpha ^{-1} \{ (T+\varepsilon )[\alpha (x)]^{T/\rho }\} ]} {d(\log x)} = \mathcal{O}(1)\) as \(x\rightarrow \infty \), for \(T,0{\lt}T{\lt}\infty \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000057">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>First we assume that \(f\) is of generalized type \(T\) with respect to the finite number \(\rho \) <i class="itshape">i.e.</i>, \(\rho {\lt}\infty \) and \(\alpha (x)\in \overline{\Omega }\). Let \(T{\lt}\infty \). Then for arbitrary \(\varepsilon {\gt}0\) and \(R{\gt}R^\prime (\varepsilon )\), </p>
<div class="displaymath" id="a0000000058">
  \begin{eqnarray*} M(R,f) {\lt}\exp [\alpha ^{-1} \{ \overline{T} [\alpha (\log R)]^\rho \} ]. \end{eqnarray*}
</div>
<p>Using (<a href="#b13">2.15</a>), we get </p>
<div class="equation" id="b20">
<p>
  <div class="equation_content">
    \begin{equation} e^p_n (f,K) / m^*_{n+1} \le A^{1/p} R(R-d)^{-(n+2)} \exp [\alpha ^{-1}\{  \overline{T} [\alpha (\log R)]^\rho \} ,\quad \overline{T}=T+\varepsilon . \label{b20}\end{equation}
  </div>
  <span class="equation_label">2.22</span>
</p>
</div>
<p>The above inequality holds for all \(n\) and \(R{\gt}R^\prime (\varepsilon )\). To minimize the right hand side of (<a href="#b20">2.22</a>) taking \(R=R(n)\) to be the unique root of the equation </p>
<div class="displaymath" id="a0000000059">
  \begin{eqnarray*} n=\tfrac {\rho }{\log R} (\alpha ^{-1} \{ \overline{T}[\alpha (\log R)]^\rho \} ), \quad n=2,3,\dots , \end{eqnarray*}
</div>
<p>or </p>
<div class="displaymath" id="a0000000060">
  \begin{eqnarray*} \log R = \alpha ^{-1} [(\tfrac {1}{\overline{T}} \alpha (\tfrac {n}{\rho }))^{\frac{1}{(\rho -1)}}] \quad \textrm{and} \quad (R-d)^{-n} \cong R^{-n}, \end{eqnarray*}
</div>
<p>substituting these values in (<a href="#b20">2.22</a>), we get </p>
<div class="displaymath" id="a0000000061">
  \begin{eqnarray*} \log (e^p_n(f,K)/m^*_{n=1}) \le -n \alpha ^{-1} [(\tfrac {1}{\overline{T}}\alpha (\tfrac {n}{\rho }))^{\frac{1}{(\rho -1)}}] + \tfrac {n}{\rho } \alpha ^{-1} [(\tfrac {1}{\overline{T}} \alpha ({n}{\rho } ))^{\frac{1}{(\rho -1)}}] \end{eqnarray*}
</div>
<p>or </p>
<div class="displaymath" id="a0000000062">
  \begin{eqnarray*} \tfrac {\rho }{\rho -1} \log (e^p_n (f,K)/m^*_{n+1})^{\frac{1}{n}} \ge \alpha ^{-1} [(\tfrac {\alpha (\frac{n}{\rho })} {\overline{T}})^{\frac{1}{(\rho -1)}}] \end{eqnarray*}
</div>
<p>or </p>
<div class="displaymath" id="a0000000063">
  \begin{eqnarray*} \overline{T} \ge \tfrac { \alpha (\frac{n}{\rho })} {[\alpha (\frac{\rho }{\rho -1} \log (e^p_n (f,K)/m^*_{n+1})^{-\frac{1}{n}})]^{\rho -1}}. \end{eqnarray*}
</div>
<p>Since \(\alpha (x)\in \overline{\Omega }\), as \(n\rightarrow \infty \), we have </p>
<div class="equation" id="b21">
<p>
  <div class="equation_content">
    \begin{equation} T\ge \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{n}{\rho })} {[\alpha (\frac{\rho }{\rho -1} \log (e^p_n (f,K)/m^*_{n+1})^{-\frac{1}{n}})]^{\rho -1}}. \label{b21} \end{equation}
  </div>
  <span class="equation_label">2.23</span>
</p>
</div>
<p>To prove the reverse inequality, we follow the method of proof of Lemma 2.3. Hence let </p>
<div class="displaymath" id="a0000000064">
  \begin{eqnarray*} \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{n}{\rho })} {[\alpha (\frac{\rho }{\rho -1} \log (e^p_n (f,K)/m^*_{n+1})^{\frac{1}{n}})]^{\rho -1}} = \sigma ^*. \end{eqnarray*}
</div>
<p>Then for a given \(\varepsilon {\gt}0\) and all \(n{\gt}n_0(\varepsilon )\), we have </p>
<div class="displaymath" id="a0000000065">
  \begin{eqnarray*} e^p_n (f,K) {\lt} m^*_{n+1} \exp \{ -\tfrac {n}{\rho } (\rho -1) \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}})\} , \quad \overline{\sigma }^* = \sigma ^*+\varepsilon . \end{eqnarray*}
</div>
<p>Now consider the function \(g(z)\) defined by the infinite series </p>
<div class="displaymath" id="a0000000066">
  \begin{eqnarray*} g(z) & =&  \sum ^\infty _{n=n_0} a^{n+1}_{n+1,R_0} z^{n+1} \exp \{  -\tfrac {n}{\rho } (\rho -1) \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}})\} , \\ & =&  \sum ^\infty _{n+n_0} b_{n+1} z^{n+1}, \textrm{ say}, \end{eqnarray*}
</div>
<p>where the sequence \(\{ a^{n+1}_{n+1,R_0}\} \) is as defined before. Since \(\alpha ^{-1} \{ [\alpha ^{-1}(\sigma \alpha (x))]^{\frac{1}{\rho }}\}  \rightarrow \infty \) as \(x\rightarrow \infty \), we get </p>
<div class="displaymath" id="a0000000067">
  \begin{eqnarray*} \limsup _{n\rightarrow \infty } [a^{n+1}_{n+1,R_0} \exp \{ \tfrac {-n}{\rho } (\rho -1) \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}})\} ]^{\frac{1}{\rho }} = 0 \end{eqnarray*}
</div>
<p>and therefore \(g(z)\) represents an entire function. Now </p>
<div class="displaymath" id="a0000000068">
  \begin{eqnarray*} \max _{z\in \Gamma _{R}} |g(z)| & =&  \sum ^\infty _{n=n_0} a^{n+1}_{n+1,R_0} R^{n+1} \exp \{ -\tfrac {n}{\rho }(\rho -1) \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })}{\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}} )\} \\ & \ge &  \sum ^\infty _{n=n_0} a^{n+1}_{n+1,R_0} R^{n+1} e^p_n (f,K) / m^*_{n+1}\\ & \ge &  A^{\tfrac {1}{p}} \| S(z) - p_{n^\prime } (z) \| _p. \end{eqnarray*}
</div>
<p>Hence if \(g(z)\) is an entire function of generalized type \(T^\prime \) with respect to the finite number \(\rho \) then from above the entire function \(g(z)\), we have </p>
<div class="equation" id="b22">
<p>
  <div class="equation_content">
    \begin{equation} T^\prime = \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{n}{\rho })} {\big[\alpha \big(\frac{\rho }{\rho -1} \log |b_n|^{-\frac{1}{n}}\big)\big]^{\rho -1}}. \label{b22} \end{equation}
  </div>
  <span class="equation_label">2.24</span>
</p>
</div>
<p>Now </p>
<div class="displaymath" id="a0000000069">
  \begin{eqnarray*} & &  |b_n|^{\frac{1}{n}} \simeq (a_{n+1,R_0}) \exp \{  \tfrac {\rho -1}{\rho } \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}})\} ,\\ & &  \tfrac {\rho }{\rho -1} \log |b_n|^{-\frac{1}{n}} \simeq \alpha ^{-1} ([\tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}]^{\frac{1}{(\rho -1)}}) (1+\mathcal{O}(1))\\ & &  [\alpha \big(\tfrac {\rho }{\rho -1} \log |b_n|^{-\frac{1}{n}}\big)]^{\rho -1} \simeq \tfrac {\alpha (\frac{n}{\rho })} {\overline{\sigma }^*}. \end{eqnarray*}
</div>
<p>Putting these values in (<a href="#b22">2.24</a>), we get \(T^\prime = \sigma ^*+\varepsilon \). As stated above we have \(T^*\ge T\). Hence we get for arbitrary \(\varepsilon {\gt}0,\sigma ^*+\varepsilon \ge T\), <i class="itshape">i.e.</i>, \(T\le \sigma ^*\). Combining this with (<a href="#b21">2.23</a>), the proof is immediate. </p>
<h1 id="a0000000070">3 MAIN RESULTS</h1>
<p>Now we shall prove our main results. </p>
<p>Following on the lines of proof of <span class="cite">
	[
	<a href="#15" >15</a>
	, 
	Th.
	
	1
	]
</span> we obtain the following inequalities for \(1\le p\le \infty \), </p>
<div class="displaymath" id="a0000000071">
  \begin{eqnarray*} \| F^{\alpha ,\beta }-H\| _p & \le &  w^{\frac{1}{p}} \| f-h\| _p, \qquad w=w(\alpha ,\beta ,p;K),\\ \| f-h\| _p & \le &  \delta ^{\frac{1}{p}} \| F^{\alpha ,\beta } - H\| _p, \qquad \delta = \delta (\alpha ,\beta ,p;K) \end{eqnarray*}
</div>
<p>for \(H\in P^{\alpha ,\beta }_{2n}\) and each \(h\in K^{-1}_{\alpha ,\beta } (H) \in p_{2n}\), \(n=0,1,2,\dots \). </p>
<p>Hence we get optimal approximates </p>
<div class="displaymath" id="c1">
  \begin{eqnarray} E^{p/2n}_{2n} (F^{\alpha ,\beta }, \textstyle \Sigma ^{\alpha ,\beta }_r) \le w^{\frac{1}{(2np)}} e^{\frac{p}{(2n)}}_{2n} (f,K), \label{c1} \end{eqnarray}
</div>
<p>and </p>
<div class="displaymath" id="c2">
  \begin{eqnarray} e^{\frac{p}{(2n)}}_{2n} (f,K) \le \delta ^{\frac{1}{(2np)}} E^{\frac{p}{(2n)}}_{2n} (F,\textstyle \Sigma ^{\alpha ,\beta }_r). \label{c2} \end{eqnarray}
</div>
<p><div class="t1_thmwrapper " id="a0000000072">
  <div class="t1_thmheading">
    <span class="t1_thmcaption">
    Theorem
    </span>
    <span class="t1_thmlabel">3.1</span>
  </div>
  <div class="t1_thmcontent">
  <p>Let \(\alpha (x)\in \overline{\Omega }\). For fixed \(p\ge 1\), let the \(F^{\alpha ,\beta }\in L^p(\Sigma ^{\alpha ,\beta }_r)\) be the restriction to \(\Sigma ^{\alpha ,\beta }_r\) of an entire GBASP function. Then \(F^{\alpha ,\beta }\) has generalized order \(\rho \) if, and only if, </p>
<div class="displaymath" id="a0000000073">
  \begin{eqnarray*} \rho = \Theta (L^*) \end{eqnarray*}
</div>
<p>where </p>
<div class="equation" id="c3">
<p>
  <div class="equation_content">
    \begin{equation} L^* = \limsup _{n\rightarrow \infty } \tfrac {\alpha (2n)} {\alpha (\log [E^p_{2n} (F^{\alpha ,\beta },E^{\alpha ,\beta }_r) / m^*_{2n+2}]^{-\frac{1}{(2n)}})} \label{c3} \end{equation}
  </div>
  <span class="equation_label">3.27</span>
</p>
</div>
<p>and \(\Theta (L^*)\) is defined as in Lemma 3. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000074">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Let \(F\in L^p(\Sigma ^{\alpha ,\beta }_r)\) be the restriction to \(\Sigma ^{\alpha ,\beta }_r\) of an entire GBASP function of generalized order \(\rho \) and let \(\varepsilon {\gt}0\) be given. From (<a href="#b10">2.10</a>), the appraisal </p>
<div class="equation" id="c4">
<p>
  <div class="equation_content">
    \begin{equation} L(f)-\varepsilon < \tfrac {\alpha (2n)} {\alpha (\log [e^p_{2n} (f,K)/m^*_{2n+2}]^{-\frac{1}{(2n)}})} < L(f) + \varepsilon , \label{c4} \end{equation}
  </div>
  <span class="equation_label">3.28</span>
</p>
</div>
<p>applies to the \(K_{\alpha ,\beta }\) associate with the lower bound holding for \(n\in n_1(\varepsilon )\), an infinite sequence of indices, and the upper bound for \(n\in n_2(\varepsilon )\), as sequence of all but a finite number of indices. From (<a href="#c2">3.26</a>), we have </p>
<div class="equation" id="c5">
<p>
  <div class="equation_content">
    \begin{equation} \tfrac {\alpha (2n)} {\alpha (\log [\delta ^{1/p} E^p_{2n} (\Sigma ^{\alpha ,\beta }_r) / m^*_{2n+2}]^{-\frac{1}{(2n)}})} > \tfrac {\alpha (2n)} {\alpha (\log [e^p_{2n} (f,K) / m^*_{2n+2}]^{-\frac{1}{(2n)}})} > L(f) - \varepsilon , \label{c5} \end{equation}
  </div>
  <span class="equation_label">3.29</span>
</p>
</div>
<p>\(n\in n_1(\varepsilon )\). For an upper estimates, (<a href="#c1">3.25</a>) gives </p>
<div class="displaymath" id="a0000000075">
  \begin{eqnarray} L(f)+\varepsilon &  {\gt}&  \tfrac {\alpha (2n)} {\alpha (\log (w^{\tfrac {1}{p}} e^p_{2n} (f,K)/m^*_{2n+2})^{-\frac{1}{(2n)}})}\nonumber \\ &  {\gt}&  \tfrac {\alpha (2n)} {\alpha (\log (E^p_{2n} (F^{\alpha ,\beta }, \Sigma ^{\alpha ,\beta }_r) / m^*_{2n+2})^{-\frac{1}{(2n)}})}, \quad n\in n_2(\varepsilon ). \label{c6} \end{eqnarray}
</div>
<p>Thus, </p>
<div class="displaymath" id="a0000000076">
  \begin{eqnarray*} L(f)-\varepsilon \le \limsup _{n\rightarrow \infty } \tfrac {\alpha (2n)} {\alpha (\log (E^p_{2n} (F^{\alpha ,\beta }, \Sigma ^{\alpha ,\beta }_r)/ m^*_{2n+2})^{-\frac{1}{(2n)}})} \le L(f)+\varepsilon . \end{eqnarray*}
</div>
<p>Hence the proof is immediate. </p>
<p><div class="t1_thmwrapper " id="a0000000077">
  <div class="t1_thmheading">
    <span class="t1_thmcaption">
    Theorem
    </span>
    <span class="t1_thmlabel">3.2</span>
  </div>
  <div class="t1_thmcontent">
  <p>Let \(\alpha (x) \in \overline{\Omega }\). For fixed \(p\ge 1\), let the \(F^{\alpha ,\beta } \in L^p(\Sigma ^{\alpha ,\beta }_r)\) be the restriction to \(\Sigma ^{\alpha ,\beta }_r\) of an entire GBASP function. Then \(F^{\alpha ,\beta }\) has generalized order \(\rho \) and finite generalized type \(T(F^{\alpha ,\beta })\) if, and only if, </p>
<div class="displaymath" id="a0000000078">
  \begin{eqnarray*} T = \limsup _{n\rightarrow \infty } \tfrac {\alpha (\frac{2n}{\rho })} {\{ \alpha [\frac{\rho }{\rho -1} \log \{ E^p_{2n} (F^{\alpha ,\beta }, \Sigma ^{\alpha ,\beta }_r)/ m^*_{2n+2}\} ^{-\frac{1}{(2n)}}]\} ^{\rho -1}} \end{eqnarray*}
</div>
<p>provided </p>
<div class="displaymath" id="a0000000079">
  \[  \frac{d(\alpha ^{-1}\{ (T+\varepsilon ) [\alpha (x)]^{T/\rho }\} )}{d(\log x)} = \mathcal{O}(1), \quad \textrm{as } x\rightarrow \infty , \textrm{for } T,0{\lt}T{\lt}\infty .  \]
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000080">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>From Lemma 4, for \(\varepsilon {\gt}0\) given, </p>
<div class="displaymath" id="a0000000081">
  \begin{eqnarray*} T-\varepsilon {\lt} \tfrac {\alpha (\frac{2n}{\rho })} {\big\{ \alpha \big[\frac{\rho }{\rho -1} \log \big(\frac{ e^p_{2n} (f,K)}{m^*_{2n+2}}\big)^{-\frac{1}{(2n)}}\big]\big\} ^{\rho -1}} {\lt}T+\varepsilon \end{eqnarray*}
</div>
<p>with the lower bound for \(n\in n_1(\varepsilon )\) and the upper bound for \(n\in n_2(\varepsilon )\). Now, using (<a href="#c2">3.26</a>) we get </p>
<div class="displaymath" id="a0000000082">
  \begin{eqnarray} T-\varepsilon &  {\lt}&  \tfrac {\alpha (\frac{2n}{\rho })} {\{ \alpha [ \frac{\rho }{\rho -1} \log \{ \frac{ e^p_{2n} (f,K)}{m^*_{2n+2}}\} ^{-\frac{1}{(2n)}}]\} ^{\rho -1}}\nonumber \\ & {\lt} & \tfrac {\alpha (\frac{2n}{\rho })} {\{ \alpha [\frac{\rho }{\rho -1} \log \{ \delta ^{1/p} E^p_{2n} (F^{\alpha ,\beta }, \Sigma ^{\alpha ,\beta }_r)/ m^*_{2n+2}\} ^{-\frac{1}{(2n)}}]\} ^{\rho -1}} \label{c7} \end{eqnarray}
</div>
<p>for \(n\in n_1(\varepsilon )\). The upper bound can be found by using (<a href="#c1">3.25</a>) </p>
<div class="displaymath" id="a0000000083">
  \begin{eqnarray} & &  \! \! \! \! \! \! \! \! \! \! \tfrac {\alpha (\frac{2n}{\rho })} {\{ \alpha [ \frac{\rho }{\rho -1} \log \{  E^p_{2n} (F^{\alpha ,\beta }, \Sigma ^{\alpha ,\beta }_r)/ m^*_{2n+2}\} ^{-\frac{1}{(2n)}}]\} ^{\rho -1}}\nonumber {\lt}\\ &  {\lt} &  \tfrac {\alpha (\frac{2n}{\rho })} {\{ \alpha [\frac{\rho }{\rho -1} \log \{ w^{1/p} e^p_{2n} (f,K) / m^*_{2n+2}\} ^{-\frac{1}{(2n)}}]\} ^{\rho -1}}\\ &  {\lt}&  T+\varepsilon \nonumber . \label{c8} \end{eqnarray}
</div>
<p>Taking limit supremum and combining (<a href="#c7">3.31</a>) and (??) we get the required result. </p>
<p><div class="ack_thmwrapper " id="a0000000084">
  <div class="ack_thmheading">
    <span class="ack_thmcaption">
    Acknwoledgments
    </span>
  </div>
  <div class="ack_thmcontent">
  <p>The author is very much thankful to the reviewers for giving fruitful comments. </p>

  </div>
</div> </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="1">1</a></dt>
  <dd><p><i class="sc">R. Askey</i>, <em>Orthogonal Polynomial and Special Functions</em>, Regional Conference Series in Applied Mathematics, SIAM Philadelphia, 1975. </p>
</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">A.A. Dovgoshei</i>, <em>Uniform polynomial approximation of entire functions on arbitrary compact sets in the complex plane</em>, Math. Zametki, <b class="bf">58</b> (1995) 3, 355–364. </p>
</dd>
  <dt><a name="3">3</a></dt>
  <dd><p><i class="sc">R.P. Gilbert</i>, <em>Function Theoretic Methods in Partial Differential Equations</em>, Math. in Science and Engineering, <b class="bf">54</b>, Academic Press, New York, 1969. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><i class="sc">G.M. Goluzin</i>, <em>Geometric Theory of Functions of one Complex Variable</em>, Nauka, Mascow, 1966. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">M. Harfaoui</i>, <em>Generalized order and best approximation of entire function in \(L_{p}\)-norm</em>, Intern. J. Maths. Math. Sci., <b class="bf">2010</b> (2010), 1–15. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><i class="sc">M. Harfaoui</i> and <i class="sc">D. Kumar</i>, <em>Best approximation in \(L_{p}\)-norm and generalized \((\alpha ,\beta )\)-growth of analytic functions</em>, Theory and Applications of Mathematics and Computer Science, <b class="bf">4</b> (2014) no. 1, 65–80. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0021-9045(81)90022-8"> <i class="sc">G.P. Kapoor</i> and <i class="sc">A. Nautiyal</i>, <em>Polynomial approximation of an entire function of slow growth</em>, J. Approx. Theory, <b class="bf">32</b> (1981), 64–75. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="8">8</a></dt>
  <dd><p><i class="sc">H.S. Kasana</i> and <i class="sc">D. Kumar</i>, <em>On approximation and interpolation of entire functions with index-pair \((p,q)\)</em>, Publicacions Mathematiques (Spain), <b class="bf">38</b> (1994), 255–267. </p>
</dd>
  <dt><a name="9">9</a></dt>
  <dd><p><i class="sc">H.H. Khan</i> and <i class="sc">R. Ali</i>, <em>Slow growth and approximation of entire solution of generalized axially symmetric Helmholtz equation</em>, Asian J. Math. Stat., <b class="bf">5</b> (2012) no. 4, 104–120. </p>
</dd>
  <dt><a name="10">10</a></dt>
  <dd><p><a href ="https://doi.org/10.1007/s11565-011-0130-8"> <i class="sc">D. Kumar</i>, <em>Generalized growth and best approximation of entire functions in \(L_{p}\)-norm in several complex variables</em>, Annali dell’Univ. di Ferrara, <b class="bf">57</b> (2011) no. 2, 353–372. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="11">11</a></dt>
  <dd><p><i class="sc">D. Kumar</i> , <em>Growth and approximation of solutions to a class of certain linear partial differential equations in \(R^{N}\)</em>, Mathematica Slovaca, <b class="bf">64</b> (2014) no. 1, 139–154. </p>
</dd>
  <dt><a name="12">12</a></dt>
  <dd><p><i class="sc">D. Kumar</i> and <i class="sc">A. Basu</i>, <em>Growth and approximation of generalized bi-axially symmetric potential</em>, J. Math. Res. Appl., <b class="bf">35</b> (2015) no. 6, 613–624. </p>
</dd>
  <dt><a name="13">13</a></dt>
  <dd><p><i class="sc">D. Kumar</i> and <i class="sc">A. Basu</i>, <em>Growth and \(L^{\delta }\)-approximation of solutions of Helmholtz equation in a finite disk</em>, J. Applied Analysis, <b class="bf">20</b> (2014), 119–128. </p>
</dd>
  <dt><a name="14">14</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0021-9045(79)90005-4"> <i class="sc">P.A. McCoy</i>, <em>Approximation of generalized biaxisymmetric potentials</em>, J. Approx. Theory, <b class="bf">25</b> (1979), 153–168. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="15">15</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/0022-247x(81)90209-2"> <i class="sc">P.A. McCoy</i>, <em>Approximation of generalized biaxially symmetric potentials on certain domains</em>, J. Math. Anal. Appl., <b class="bf">82</b> (1981), 463–469. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="16">16</a></dt>
  <dd><p><i class="sc">M.N. Sheremeta</i>, <em>On the connection between the growth of zero-order functions which are entire or analytic in a circle and coefficients of their power expansions</em>, Izv. Vuzov. Mat., (1968) no. 6, 115–121. </p>
</dd>
  <dt><a name="17">17</a></dt>
  <dd><p><a href ="https://doi.org/10.1090/trans2/088/11"> <i class="sc">M.N. Sheremeta</i>, <em>On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion</em>, Trans. Amer. Math. Soc., <b class="bf">88</b> (1970), 291–301. <img src="img-0001.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="18">18</a></dt>
  <dd><p><i class="sc">G. Szegö</i>, <em>Orthogonal Polynomials</em>, Colloquium Publications, 23, Amer. Math. Soc., Providence, R.I. 1967. </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>