\(^\ast \)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: d_kumar001@rediffmail.com

1 INTRODUCTION

Let \(F^{\alpha ,\beta }\) be a real valued regular solution to the generalized biaxisymmetric potential equation

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

such that

where the \(P_n^{\alpha ,\beta }\) are Jacobi polynomials [ 1 ] , [ 18 ] .

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

and the space \(l^p(K)\) of associated functions

continuous on \(K\) with finite \(p\)-norm. Following McCoy [ 14 ] for Koornwinder’s integral for Jacobi polynomials and the inverse operator have been defined as:

and

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 [ 3 ] .

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 [ 4 , p.285 ] we put

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

Then we have [ 4 , pp. 287–289 ] ,

where \(\widetilde{d}\) is the transfinite diameter of \(K\).

Using the Koornwinder’s integral and inverse operator the information concerning the approximation and growth of analytic functions can be transfer to GBASP \(F^{\alpha ,\beta }\).

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

and

It is the Bernstein limits of the optimal approximates,

and

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 \(P^{\alpha ,\beta }_{2n}\).

Several authors such as Harfaoui [ 5 ] , Kumar [ 10 ] , Harfaoui and Kumar [ 6 ] 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 [ 12 ] , [ 13 ] , Kumar [ 11 ] , Khan and Ali [ 9 ] .

McCoy [ 15 , Th. 2 ] 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 \(\Sigma ^{\alpha ,\beta }_r\).

2 GENERALIZED ORDER AND GENERALIZED TYPE

Seremeta [ 17 ] defined the generalized order and generalized type with the help of general functions as follows.

Let \(L^0\) denote the class of functions \(h\) satisfying the following conditions:

1. \(\phi (x)\) is defined on \([a,\infty )\) and is positive, strictly increasing, differentiable and tends to \(\infty \) as \(x\rightarrow \infty \).

2. It holds

   \begin{eqnarray*} \lim _{x\rightarrow \infty } \tfrac {\phi \left\{ \left(1+\frac{1}{\varphi (x)}\right) (x)\right\} } {\phi (x)} = 1 \end{eqnarray*}

   for every function \(\varphi (x)\) such that \(\varphi (x)\rightarrow \infty \), as \(x\rightarrow \infty \).

   Let \(\Delta \) denote the class of functions \(\phi \) satisfying condition (i) and the following:

3. It holds

   \begin{eqnarray*} \lim _{x\rightarrow \infty } \tfrac {\phi (c x)} {\phi (x)} = 1 \end{eqnarray*}

   for every \(c{\gt}0\), that is, \(\phi (x)\) is slowly increasing.

   For an entire function \(f(z)\) and functions \(\alpha (x)\in \Delta ,\beta (x)\in L^0\), Seremeta [ 10 , Th. 1 ] proved that

   \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*}

   Further, for \(\alpha (x)\in L^0, \beta ^{-1}(x) \in L^0\) and \(\gamma (x) \in L^0\), we have

   \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*}

   where \(0{\lt}\rho {\lt}\infty \) is a fixed number.

   The above relations were obtained under certain conditions which do not hold if \(\alpha = \beta \). In 1968, Seremeta [ 16 ] 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:

   \begin{eqnarray*} \rho _\alpha = \limsup _{n\rightarrow \infty } \frac{\alpha \left(\log \log M(r,f)\right)}{\alpha (\log \log r)}, \end{eqnarray*}

   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

   \begin{eqnarray*} 0\le \tfrac {dF(x,c)}{dx} \le A\big(\exp (F(x,c))^B\big) \end{eqnarray*}

   is realized for any \(x\ge x_1\), where \(A\) and \(B\) are some constants \((0{\lt}A,B{\lt}\infty )\), then we obtain

   \begin{eqnarray*} \rho _\alpha = \max (\rho ^\prime _\alpha ,1).\end{eqnarray*}

   Here

   \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*}

   To refine this scale, Kapoor and Nautiyal [ 7 ] introduced a new class of functions as follows: A function \(\phi (t)\in \Omega \) if \(\phi (t)\) satisfies (ii) and:

4. There exists a function \(\delta (t) \in \Delta \) and \(t_0,K_1\) and \(K_2\) such that for all \(t{\gt}t_0\)

   \begin{eqnarray*} 0{\lt}K_1 \le \tfrac {d(\phi (t))} {d(\delta (\log t))} \le K_2 {\lt}\infty . \end{eqnarray*}

   Further a function \(\phi (t)\in \overline{\Omega }\) if \(\phi (t)\) satisfies (ii) and

5. \begin{eqnarray*} \lim _{t\rightarrow \infty } \tfrac {d(\phi (t))} {d(\log (t))} = K, \quad 0{\lt}K{\lt}\infty .\end{eqnarray*}

   Kapoor and Nautiyal [ 7 , p. 66 ] 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 [ 7 , p. 66 ] , for entire GBASP \(F^{\alpha ,\beta }\) and associate we define the generalized order and generalized type as

   \begin{equation} \rho = \rho (\alpha ,\alpha ,f) = \limsup _{r\rightarrow \infty } \tfrac {\alpha (\log M(R,f))} {\alpha (\log r)} \label{b1}\end{equation}

   2.1

   \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}

   2.2

   \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}

   2.3

   \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}

   2.4

   where

   \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*}

   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 \(r{\gt}1\).

McCoy [ 14 ] proved the following result:

Now we prove the following Lemmas:

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

The inverse relation

leads to the inequality

where

However, for \(z=\varepsilon r e^{i\theta } (\varepsilon \) real),

it gives

Using inequalities (2.5), (2.6) and definitions (2.1)–(2.4), the proof is completed.

Using the definitions of generalized order and generalized type of entire GBASP and proof proceeds on the lines of Lemma 1 [ 6 ] .

Let \(d=\max \{ |z|;z\in K\} \). From the definition of \(e^p_n(f,K)\), since \(h\in p_n\), we have

where \(A\) is the area of \(K\).

Now using Lemma 2 [ 2 , p. 923 ] , it has been shown that for \(R{\gt}d\),

Using (2.14) in (2.13) we get

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 \(R{\gt}R_0(\varepsilon ), R(\varepsilon )\in (0,\infty )\),

In view of (2.15) and (2.16) we have

Since \(d\) is finite and fixed and the above inequality holds for all \(R{\gt}R_0 (\varepsilon )\), we can choose

Substituting this value of \(R\) in (2.17), we obtain

since \(F\big(n,\tfrac {1}{\overline{\rho }-1}\big)\rightarrow \infty \) as \(n\rightarrow \infty \). Hence

Since \(\alpha (x)\in \overline{\Omega }\), as \(n\rightarrow \infty \), we have

In order to prove reverse inequality, let us put

Suppose \(L(f){\lt}\infty \). Then for given \(\varepsilon {\gt} 0\) and all \(n{\gt}n_0(\varepsilon )\), we have

Now we consider the function

where \(R_0{\gt}d\) and

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

In view of proof of [ 2 , Th. 1, p. 924 ] , it can be easily seen that the series (2.20) is uniformly convergent on \(\Gamma _{R}\) for any arbitrary \(R{\gt}0\). Thus the sum represent an entire function. Now

On \(K\subseteq \mathbb {C}\) we have the inequality

Dovgoshei [ 2 , p. 924 ] shown that

For \(1\le p\le \infty \), we have

leading to the relation

From the last inequality, using [ 7 , Th. 4 ] , 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 \(f\).

If \(\rho _{1}\) denotes the generalized order of \(h^*(z)\) then

Since

Hence

Since \(\alpha (x)\in \overline{\Omega }\), we finally get

Since \(\varepsilon \) was arbitrary, we get \(\rho _1\ge \rho (f)\). Combining this with (2.18) we get (2.9).

First we assume that \(f\) is of generalized type \(T\) with respect to the finite number \(\rho \) i.e., \(\rho {\lt}\infty \) and \(\alpha (x)\in \overline{\Omega }\). Let \(T{\lt}\infty \). Then for arbitrary \(\varepsilon {\gt}0\) and \(R{\gt}R^\prime (\varepsilon )\),

Using (2.15), we get

The above inequality holds for all \(n\) and \(R{\gt}R^\prime (\varepsilon )\). To minimize the right hand side of (2.22) taking \(R=R(n)\) to be the unique root of the equation

or

substituting these values in (2.22), we get

or

or

Since \(\alpha (x)\in \overline{\Omega }\), as \(n\rightarrow \infty \), we have

To prove the reverse inequality, we follow the method of proof of Lemma 2.3. Hence let

Then for a given \(\varepsilon {\gt}0\) and all \(n{\gt}n_0(\varepsilon )\), we have

Now consider the function \(g(z)\) defined by the infinite series

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

and therefore \(g(z)\) represents an entire function. Now

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

Now

Putting these values in (2.24), 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.e., \(T\le \sigma ^*\). Combining this with (2.23), the proof is immediate.

3 MAIN RESULTS

Now we shall prove our main results.

Following on the lines of proof of [ 15 , Th. 1 ] we obtain the following inequalities for \(1\le p\le \infty \),

for \(H\in P^{\alpha ,\beta }_{2n}\) and each \(h\in K^{-1}_{\alpha ,\beta } (H) \in p_{2n}\), \(n=0,1,2,\dots \).

Hence we get optimal approximates

and

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 (2.10), the appraisal

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 (3.26), we have

\(n\in n_1(\varepsilon )\). For an upper estimates, (3.25) gives

Thus,

Hence the proof is immediate.

From Lemma 4, for \(\varepsilon {\gt}0\) given,

with the lower bound for \(n\in n_1(\varepsilon )\) and the upper bound for \(n\in n_2(\varepsilon )\). Now, using (3.26) we get

for \(n\in n_1(\varepsilon )\). The upper bound can be found by using (3.25)

Taking limit supremum and combining (3.31) and (??) we get the required result.