\(^{1}\)Ankara University, Faculty of Science, Department of Mathematics, 06100, Tandoan, Ankara, Turkey, e-mail: tunca@science.ankara.edu.tr.

\(^{2}\)Abant Ýzzet Baysal University, Faculty of Arts and Sciences, Department of Mathematics, 14280, Bolu, Turkey, e-mail: erencina@hotmail.com.

1 Introduction

In [ 5 ] , Butzer introduced the Kantorovich variant of the Szasz-Mirakyan operators as

for each \(n\in \mathbb {N}\) and \(f\in L^{1}(0,\infty )\), the space of integrable functions on unbounded interval \([0,\infty ).\) The \(L^{p}(1{\lt}p{\lt}\infty )\) saturation and inverse theorems for the Szasz-Mirakyan-Kantorovich operators were studied by Totik in [ 19 ] . Bezier variant of the Szasz-Mirakyan-Kantorovich operators were introduced by Gupta, Vasishtha and Gupta in [ 8 ] , where the rate of convergence of these operators for functions of bounded variation was measured. Moreover, approximation properties of these operators for locally bounded functions have been investigated recently by Gupta and Xiao-Ming Zeng in [ 9 ] . In [ 7 ] , Duman, Özarslan and Della Vecchia constructed a modified Szasz-Mirakyan-Kantorovich operators and introduced a better error estimation and a Voronovskaya type theorem. Another interesting modification of Szasz-Mirakyan-Kantorovich operators has been carried out by Nowak and Sikorska-Nowak in [ 15 ] , where the operators are defined for Denjoy-Perron integrable functions \(f\). The authors have obtained an estimate for the rate of pointwise convergence for such operators at the Lebesgue Denjoy points of \(f\).

In \(1996\), Phillips’ generalization of Bernstein polynomials by using \(q\)-integers [ 16 ] has brought up a new point of view to the theory of approximation by linear positive operators. There are numerous works dealing with this approach. Some are in [ 2 ] , [ 3 ] , [ 4 ] , [ 6 ] , [ 11 ] , [ 12 ] , [ 13 ] , [ 20 ] , [ 21 ] and references therein.

In this work, we consider a Kantorovich type generalization of \(q\)-Szasz-Mirakyan operators, introduced by Aral in [ 4 ] with the help of the Riemann type \(q\)-integral defined in [ 14 ] and prove a Voronovskaya type theorem. Observe that a different form of \(q\)-Szasz-Mirakyan-Kantorovich operators has been studied in [ 13 ] by Mahmudov and Gupta.

2 Notation and construction of the operators

In this section, we first mention some well-known definitions of \(q\)-calculus. Next introduce the \(q\)-Szasz-Mirakyan-Kantorovich operators via Riemann type \(q\)-integral.

Let \(q{\gt}0\) be a fixed real number and \(n\) be a nonnegative integer. The respective definitions of the \(q\)-integer \([n]_{q}\) and the \(q\)-factorial \([n]_{q}!\) are (see [ 1 ] )

The two \(q\)-analogues of the exponential function \({\rm e}^{x}\) are given by

and

where \(\displaystyle (x;q)_{\infty }=\prod ^{\infty }_{k=1}(1-xq^{k-1})\) (see [ 10 ] ). We know that

and

In [ 4 ] , Aral defined the \(q\)-Szasz-Mirakyan operators as follows:

where \(n\in \mathbb {N}\), \(0{\lt}q{\lt}1\), \(f\in C[0,\infty )\) and \((b_{n})\) is an increasing sequence of positive numbers such that \(\displaystyle \lim _{n\rightarrow \infty }b_{n}=\infty .\) He proved a Voronovskaya type theorem using \(q\)-derivatives and also obtained convergence properties of these operators and their derivatives. Note that the operators defined by \((2.1)\) are the \(q\)-extension of the Szasz-Chlodovsky operators introduced in [ 18 ] by Stypinski. From [ 4 ] we have

where \(e_{m}(t):=t^{m},\) \(m=0,1,2,3,4.\)

Now assume that \(0{\lt}a{\lt}b\), \(0{\lt}q{\lt}1\) and \(f\) is a real-valued function. The \(q\)-Jackson integral of \(f\) over the interval \([0,b]\) and over a general interval \([a,b]\) are defined by (see [ 10 ] )

and

respectively, provided the series converge. It is easily seen that the inequality \(f(x)\geq 0\) for all \(x\in [a,b]\) does not ensure that \(\int ^{b}_{a}f(x){\rm d}_{q}x\geq 0\). This fact may not be convenient in order to guarantee the positivity of our operator. On the other hand, since the \(q\)-Jackson integral contains two infinite sums, in obtaining the \(q\)-analogues of some well-known integral inequalities which are used to estimate the order of approximation of linear positive operators involving \(q\)-Jackson integral, some problems are encountered. Because of these circumstances, we consider the Riemann type \(q\)-integral introduced in [ 14 ]

to construct a Kantorovich variant of the operator defined by \((2.1).\)

With this motivation, we now introduce the Kantorovich type generalization of the \(q\)-Szasz-Mirakyan operators as follows:

where \(x\in \big[0,\tfrac {b_{n}}{(1-q)[n]_{q}}\big)\), \(n\in \mathbb {N}\), \(0{\lt}q{\lt}1\), \(f\) is Riemann type \(q\)-integrable on \(\big[\tfrac {[k]_{q}b_{n}}{[n]_{q}},\tfrac {[k+1]_{q}b_{n}}{[n]_{q}}\big]\) and \((b_{n})\) is an increasing sequence of positive numbers having the property \(\displaystyle \lim _{n\rightarrow \infty }b_{n}=\infty .\)

3 Auxiliary results

In this section, we give some lemmas which will be used to prove our main result.

Interchanging the order of summations over \(i\) and \(l\) and then taking \(s=i-l\) in the obtained result, one may write

where each \(C_{m,l}(q,b_{n})\) is given by \((3.1).\) Thus the proof is completed.

Now, by means of the Lemma 1 we can evaluate \(K_{n,q}(e_{m};x).\)

In the light of Corollary 1 and the linearity of \(K_{n,q}\), we now introduce the following Lemma without proof.

Now, with the help of the Lemma 3, we can evaluate the following limits which will be essential in the proof of the main theorem.

Now from \((3.8),\) we can write

and

Since \(\displaystyle \lim _{n\rightarrow \infty }q_{n}=1\), \(\displaystyle \lim _{n\rightarrow \infty }b_{n}=\infty \) and \(\displaystyle \lim _{n\rightarrow \infty }\tfrac {b_{n}}{[n]_{q_{n}}}=0\) this gives

Similarly it can be shown that \((3.12)\) holds.

4 Main result

Now, we can establish a Voronovskaya type result for the operator \(K_{n,q}(f;x)\) defined by \((2.4)\).