### Article

Kyungpook Mathematical Journal 2020; 60(1): 126-132

**Published online** March 31, 2020

Copyright © Kyungpook Mathematical Journal.

### Hardy Spaces of Certain Convolution Operator

Rajbala and Jugal Kishore Prajapat∗

Department of Mathematics, Central University of Rajasthan, Ajmer-305817, Rajasthan, India

e-mail : rajbalachoudary9@gmail.com and jkprajapat@gmail.com

**Received**: January 14, 2018; **Revised**: March 20, 2019; **Accepted**: April 9, 2019

In this article, we determine sufficient conditions on the parameters of a generalized convolution operator to ensure that it belongs to the Hardy space and to the space of bounded analytic functions. We exhibit the utility of these results by deducing several interesting examples.

**Keywords**: analytic functions, bounded analytic functions, Hardy spaces.

Let ℋ denote the class of analytic functions in the open unit disk , and be the subclass of ℋ consisting of normalized analytic functions of the form

For the analytic functions

Further, for functions , given by _{1} and _{2} is defined by

Let ℛ(

Let ℋ^{∞} denote the space of all bounded analytic functions in . This is Banach algebra with respect to the norm . For

The function ^{p}_{p}

(see [3, p. 2]). For 1 ≤ ^{p}

(see [3, p. 23]). Following are two widely known results (see [8]) for the Hardy space ℋ^{p}

In [10], Ponnusamy studied the Hardy space of hypergeometric functions. Further, Baricz [1] obtained the conditions for the generalized Bessel functions such that it belongs to Hardy space and Yagmur and Orhan [16] studied the same problem for the generalized Struve functions.

In [6] (see also [7]), Ibrahim studied the following generalized fractional integral operator in the complex plane ℂ:

where the function ^{μ}^{+1} −^{μ}^{+1})^{α}^{−1} is removed by requiring log(^{μ}^{+1} − ^{μ}^{+1}) to be real when (^{μ}^{+1}−^{μ}^{+1}) > 0. We observe that, if we take ^{+} in

For

Note that

Corresponding to fractional integral operator ^{α,μ}f

In order to derive our main results, we recall here the following lemmas:

### Lemma 2.1.([15])

### Lemma 2.2.([14])

Our first main result is given by Theorem 2.3 below.

### Theorem 2.3

**Proof**

From the definition of operator

By hypothesis

Since _{n}

where

Evidently,

Thus Γ̃(

Hence

which shows that 0 <

Now, we shall show that lim_{n}_{→}_{∞}^{1/}^{n}

This implies that the series in

Also, using the known result [3, Theorem 3.11],

### Remark 2.4

If we consider ^{∞}, but as per the Theorem 2.3, function

### Theorem 2.5

**Proof**

The inequality

which in view of

### Theorem 2.6

**Proof**

Let

### Acknowledgements

The authors wish to thank the referee for her/his valuable suggestions, which have improved the paper. The work was supported by the University Grant Commission, India (Project Number: MRP-MAJOR-MATH-2013-19114).

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