Abstract

Inequalities play a fundamental role in many branches of mathematics and particularly in real analysis. By using inequalities, we can find extrema, point of inflection, and monotonic behavior of real functions. Subordination and quasisubordination are important tools used in complex analysis as an alternate of inequalities. In this article, we introduce and systematically study certain new classes of meromorphic functions using quasisubordination and Bessel function. We explore various inequalities related with the famous Fekete-Szego inequality. We also point out a number of important corollaries.

1. Introduction

A complex valued function is said to be meromorphic if it has poles as its only singularities. Let denotes the class of all of meromorphic functions which has a simple pole at and has Laurent series expansion of the form: which are analytic in the punctured open unit disc as open unit disc

Here, we are listing some important subclasses of meromorphic functions which will be used in our subequal work. In 1936, Robertson [1] introduced the classes of meromorphic starlike and meromorphic convex functions of order By , we mean the subclass of consisting of all meromorphic starlike functions of order. Analytically,

A closely related class of meromorphic convex functions of order is denoted by and defined as

In 1952, W. Kaplan [2] introduced and studied an important class of analytic functions known as close-to-convex functions in the open unit disc . A function belongs to , is in class , of meromorphic close-to-convex functions of order and type if there exist, and

Let and having series representation of the form

Then, the convolution of and as denoted by is defined as where is given by (1).

A function is subordinate to in and written as if there exists a Schwarz function , which is holomorphic in with , such that Let be an analytic function with positive real part on satisfies and which maps which is star shape with respect to also symmetric with respect to the real axis. We denote be the class of function for which The class was introduced and studied by Silverman et al. [3] (see also [4]). The class is a special case of the class when

Robertson [5] gave the idea of quasisubordination. For any two functions and , holomorphic in , the function is said to be quasi-subordinate to the function written as if there exists two holomorphic functions and , with , is holomorphic in and such that . In particular, if , then quasisubordination reduces to subordination. Furthermore, if the quasisubordination becomes the majorization, (see [6]), which implies

For recent work on meromorphic functions, we refer [7–18].

Motivated from the above cited work, we introduce the following subclasses of meromorphic functions. Throughout in this paper, we shall assume , , , , and be an analytic function with positive real part on that satisfies and which maps which is star shape with respect to , and also symmetric with respect to the real axis unless otherwise mentioned.

Definition 1. Let be the class of functions and satisfy

The abovementioned class is the meromorphic analogue of the class introduced and studied by Mohd and Darus [19]. For , the class was studied by Zayed et al. [20].

Definition 2. Let be the subclass of consisting of all functions for which their exist and satisfy

For and , the class was studied by Zayed et al. [20].

In this paper, we obtain the Fekete-Szego inequality for meromorphic functions belonging to above defined classes. Let denote the class of functions of the form satisfying , for . For more details, see [21–23]. To prove our main results, we need the following lemma.

Lemma 3 [24]. If , then for any complex number , the result is sharp for the functions given by

2. Main Results

In this section, we explore certain Fekete-Szego-related inequalities for the class and .

Theorem 4. Let , and if given by (1) be in the class , and is a complex number, then

The inequalities are sharp for or .

Proof. Let , then there exist analytic functions and , with , , and such that Taking first and second derivative of (1), and use in the left hand side of above equation, we obtain then implies then which implies Comparing (13) and (16), we get Thus, Since is analytic and bounded in (see [25]), so we have By using this fact and the well-known inequality we get

Corollary 5. For and in Theorem 4, we obtain the result by Silverman et al. [3] (see Theorem 4).

For and repeating steps of Theorem 4, we obtain the following corollary.

Corollary 6. Let satisfies then for any complex number,

Theorem 7. Let , and if , , given by (1) and (5) be in the class , and is a complex number, then

The inequalities is sharp for or .

Proof. Let , , then there exist analytic functions and , with , , and such that Taking first derivative of (1) and (5), and use in the left hand side of above equation, we obtain then implies which implies Comparing (26) and (28), we get Thus, Since is analytic and bounded in (see [25]), we have By using this fact and the well-known inequality , we get

Corollary 8. For , , and in Theorem 7, we obtain the result by Silverman et al. [3] (see Theorem 7). For and repeating steps of Theorem 7, we obtain the following corollary.

Corollary 9. Let and satisfy then for any complex number ,

Let us consider the second order linear homogenous differential equation (see, Baricz [26])

The function is known as generalized Bessel’s function of first kind and is the solution of differential equation given in (35). If we denote where , , and are positive real numbers. The operator is a meromorphic analogue introduced by Deniz [27] (see also Baricz et al. [28]) for analytic functions. In terms of convolution, is given by

The operator was introduced and studied by Mostafa et al. [29]. For more details, see [30, 31] and references cited therein. Motivated from the above cited work, we introduce the following classes of meromorphic functions.

Definition 10. A function given by (1) is said to belong to the class if

For , the class was studied by Zayed et al. [20].

Definition 11. Let be the subclasses of consisting of all functions for which there exist , and

For and , the class was studied by Zayed et al. [20].

Theorem 12. Let , and if given by (1) be in the class , and is a complex number, then

Proof. Let , then there exist analytic functions and , with , , and such that Taking first and second derivative of (38), in use of the left side of the above equation, we obtain thus implies which implies Comparing (44) and (46), we get Thus, Since is analytic and bounded in (see [25]), we have ,
By using this fact and the well-known inequality , we get

We have thus completed the proof of Theorem 12.

For and repeating steps of Theorem 12, we obtain the following corollary.

Corollary 13. Let satisfies

Then, for any complex number ,

Theorem 14. Let , and if and given by (1) and (5) be in the class and is a complex number, then

Proof. Let and then there exist analytic functions and , with , , and such that Taking first derivative of (5) and (38) in use of the left side of above equation, we have and this implies which implies Comparing (55) and (57), we get Thus, Since is analytic and bounded in (see [25]), we have By using this fact and the well-known inequality, we get

We have thus completed the proof of Theorem 14.

For and repeating steps of Theorem 14, we obtain the following corollary.

Corollary 15. Let and satisfy then for any complex number ,

4. Conclusion

In our present investigation, we have defined and systematically studied the famous Fekete-szego inequality for some subclass of meromorphic functions by using quasisubordination. It is important to mention that certain results in the literature, for example [3, 19, 20], are special cases of the results obtained by us.

Data Availability

No data is used.

Conflicts of Interest

The authors declare that they have no conflict of interest.

Authors’ Contributions

SH came with the main thoughts and helped to draft the manuscript. SGAS and AR proved the main theorems. ZS and MD revised the paper. All authors read and approved the final manuscript.

Funding

The fifth author is supported by GUP-2019-032.