Journal of Function Spaces
Volume 2018 (2018), Article ID 6327132, 7 pages
https://doi.org/10.1155/2018/6327132

Research Article

Convexity of Certain Integral Operators Defined by Struve Functions

Shahid Mahmood,¹ Sana Mahroz,² Ayesha Rafiq,³ Sarfraz Nawaz Malik,² and Mohsan Raza⁴

¹Department of Mechanical Engineering, Sarhad University of Science and Information Technology, Ring Road, Peshawar, Pakistan
²COMSATS Institute of Information Technology, Wah Cantt, Pakistan
³Institute of Space Technology, Islamabad, Pakistan
⁴Department of Mathematics, Government College University, Faisalabad, Pakistan

Correspondence should be addressed to Sarfraz Nawaz Malik; snmalik110@yahoo.com

Received 4 December 2017; Accepted 28 January 2018; Published 1 March 2018

Academic Editor: Henryk Hudzik

Copyright © 2018 Shahid Mahmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This article deals with some functional inequalities involving Struve function, generalized Struve function, and modified Struve functions. We aim to find the convexity of the integral operator defined by Struve function, generalized Struve function, and modified Struve functions.

1. Introduction

We denote by the class of functions which are analytic in the open unit disc and of the form

Let denote the class of all functions in which are univalent in . Also let and be the subclasses of consisting of all functions which map onto a star shaped with respect to origin and convex domains, respectively. These classes are defined as

Ozaki [1] showed that if a function is analytic in and of the form (1) such that and if either

or

then is univalent and convex in at least one direction in . It shows that the constants and are, in a certain sense, the best possible constants. In [2], for , consider the class consisting of locally univalent functions that satisfy the condition for That is,

It can be seen that if and only if . Since whenever , it readily follows that the class is in , for , so in particular, the functions in are univalent functions. For some more detail about these conditions, see [3, 4].

Now we consider the second-order inhomogeneous differential equation

The solution of the homogeneous part is Bessel functions of order , where is real or complex number. For some more details, see [5]. The particular solution of the inhomogeneous equation defined in (6) is called the Struve function of order . It is defined as

Now we consider the differential equation

Equation (8) differs from (6) in the coefficients of . Its particular solution is called the modified Struve functions of order and is given by

Again consider the second-order in-homogenous differential equation

where Equation (10) generalizes (6) and (8). In particular for , we obtain (6) and for , we obtain (8). Its particular solution has the series form

and is called the generalized Struve function of order . This series is convergent everywhere. We take the transformation

where and , where denotes the Gamma function. The function is analytic in the whole complex plane and satisfies the differential equation

The function unifies the Struve functions and modified Struve functions.

The function was introduced and studied by Orhan and Yagmur [6] and further investigated by [7–9]. In last few years, many mathematicians have set the univalence criteria of several those integral operators which preserve the class . By using a variety of different analytic techniques, operators, and special functions, several authors have studied univalence criterion. Recently Din et al. [10] studied the univalence of integral operators involving generalized Struve functions. These operators are defined as follows:

In this paper, our main aim is to study the convexity and univalence of the integral operators

Applications of Struve functions occur in water wave and surface wave problems, unsteady aerodynamics, resistive MHD instability theory, and optical diffraction. More recently, Struve functions appeared in particle quantum dynamic studies of spin decoherence and nanotubes electrodynamics, potential theory, and optics. For more details, we refer to see [11].

Lemma 1 (see [6]). If and , are so constrained that , then the function satisfies the following inequalities:

(i),(ii)

2. Main Results

In this section, we find the convexity of these integral operators defined by generalized Struve functions, by using above lemma and the following inequalities. We also use Ozaki’s condition for the univalence of these operators.

2.1. Convexity Criteria for Integral Operators

Theorem 2. Let and be so constrained that , where . Consider the function , defined as

Also suppose that and be positive real number and that satisfies the inequality

then defined in (15) is in class and

Proof. It is clear that as it satisfies the condition

Differentiating (15), we obtain

Clearly

Differentiating logarithmically and after simple calculations, it follows that

Using inequality (i) of Lemma 1, it is easy to see that

so

By putting the above inequality in (23), we have

For and , we consider that the function

defined by

is decreasing , and therefore

Hence

which completes the proof that

Theorem 3. Let and be so constrained that where . Consider the function , defined as

Also suppose that and be positive real number and that satisfies the inequality

then defined in (16) is in class , where

Proof. Differentiating (16), we obtain

Clearly

Differentiating logarithmically and after simple calculations, we obtain

By using inequality (ii) of Lemma 1, we have

so

Using above relation in (36), we have

For and ,we consider that the function

defined by

is decreasing , and therefore

Hence

We have which completes the proof that

3. Some Special Cases of Struve Functions and Modified Struve Functions

3.1. Struve Functions

By setting in (11), we obtain the Struve function of first kind of order , denoted by defined in (7). Let be defined as

We observe that

Corollary 4. Let , where Consider the function defined by

Also let and be positive real numbers. Suppose that these numbers satisfy the following inequality.

then the function defined as

is in class , where

In particular
(i) First at , the function

is in class for , where

(ii) For , then the function

is in class for , where
(iii) For , the function

is in class for , where .

Corollary 5. Let where Consider the function defined by (46) and and be positive real numbers such that these numbers satisfy the inequality

then the function defined as

is in class , where and

In particular, one has the following:
(i) For , then

is in class for , where
(ii) For , the function

is in class for , where .
(iii) For , the function

is in class for , where

3.2. Modified Struve Functions

We obtain the modified Struve function of first kind of order , denoted by , defined by (9), by putting in (11). Define a function by

We observe that

By making use of this particular value, we have the following assertions.

Corollary 6. Let where Consider the function defined by

and and be positive real numbers. Suppose that these numbers satisfy the inequality

then the function defined as

is in class , where

(i) In particular for , the integral operator

is in class for , where

Corollary 7. Let , where Consider the function defined by

, and be positive real numbers. Suppose that these numbers satisfy the relation

then the function defined as

is , where

In particular for , the function

is in class for , where

4. Locally Univalence Criteria

Now, we find the locally univalent criteria of the integral operators involving generalized Struve functions defined in (15), (16).

Theorem 8. Let and such that , where . Consider the function , defined as

Also suppose that and be positive real number and that satisfies the inequality

then defined in (16) is in class , where

Proof. Differentiating (16), we obtain

Differentiating logarithmically, we obtain

By using (ii) of Lemma 1, it follows that

Therefore, we have

For and , we consider the function

defined by

is decreasing ; therefore

This implies that

Let

Hence

Here , which completes the proof.

Theorem 9. Let and be such that , where . Consider the function , defined as

Also suppose that and are positive real number and that satisfies the inequality

then defined in (15) is in class , where

Proof. Differentiating (15), we have

Differentiating logarithmically, we obtain

Now consider (i) of Lemma 1, we obtain

Therefore

For and , we consider the function

defined by

therefore

This implies that

Let

Hence

Here , which completes the proof.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge the heads of their institutions for their supportive role and research facilities. This research is conducted under the Project of Higher Education Commission of Pakistan (Project Ref: 5689/Punjab/NRPU/R&D/2016) and is partially supported by Sarhad University of Science and Information Technology, Peshawar, Pakistan.

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