Application of Quasisubordination to Certain Classes of Meromorphic Functions

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.


Introduction
A complex valued function is said to be meromorphic if it has poles as its only singularities. Let ∑ 1 denotes the class of all of meromorphic functions which has a simple pole at ω = 0 and has Laurent series expansion of the form: which are analytic in the punctured open unit disc U * = fω : ω ∈ C and 0 < jωj < 1g = U − f0g, as open unit disc U = U * ∪ f0g: 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 α: By ∑ MS ðαÞ, we mean the subclass of ∑ 1 consisting of all meromorphic starlike functions of order α. Analytically, A closely related class of meromorphic convex functions of order α is denoted by ∑ MC ðαÞ 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 U. A function λ belongs to ∑ 1 , is in class ∑ MC ðα, βÞ, of meromorphic close-to-convex functions of order α and type β if there exist δðωÞ ∈ ∑ MS ðβÞ, and Let δðωÞ ∈ ∑ 1 and having series representation of the form Then, the convolution of λ and δ as denoted by λ * δ is defined as where λ is given by (1). A function λ is subordinate to δ in U * and written as λ ðωÞ ≺ δðωÞ, if there exists a Schwarz function kðωÞ, which is holomorphic in U with kð0Þ = 0, such that λðωÞ = δðkðωÞÞ: Let ϕðωÞ be an analytic function with positive real part on U satisfies ϕð0Þ = 1 and ϕ ′ ð0Þ > 0 which maps U which is star shape with respect to ω = 1, also symmetric with respect to the real axis. We denote ∑ðϕÞ be the class of function λ∈∑ 1 for which −ωλ′ðωÞ/λðωÞ ≺ ϕðωÞ, ð ω ∈ U * Þ: The class ∑ðϕÞ was introduced and studied by Silverman et al. [3] (see also [4]). The class ∑ðαÞ is a special case of the class ∑ðϕÞ when Robertson [5] gave the idea of quasisubordination. For any two functions λðωÞ and δðωÞ, holomorphic in U, the function λðωÞ is said to be quasi-subordinate to the function δðωÞ written as λðωÞ≺ q δðωÞ, if there exists two holomorphic functions λðωÞ and φðωÞ, with jφðωÞj ≤ 1, λðωÞ/φðωÞ is holomorphic in U * and such that λðωÞ = φðωÞδðkðωÞÞ. In particular, if φðωÞ = 1, then quasisubordination reduces to subordination. Furthermore, if kðωÞ = 1, the quasisubordination becomes the majorization, (see [6]), which implies For recent work on meromorphic functions, we refer [7][8][9][10][11][12][13][14][15][16][17][18].
Motivated from the above cited work, we introduce the following subclasses of meromorphic functions. Throughout in this paper, we shall assume 0 ≤ γ < 1, γ ≠ 1/2, 0 ≤ η < 1, ω ∈ U * , λ, δ ∈ ∑ 1 , and ϕðωÞ be an analytic function with positive real part on U that satisfies ϕð0Þ = 1 and ϕ′ð0Þ > 0 which maps U which is star shape with respect to ω = 1, and also symmetric with respect to the real axis unless otherwise mentioned.
Definition 1. Let ∑ MC q ðϕ, γÞ be the class of functions λðωÞ ∈ ∑ 1 and satisfy The abovementioned class ∑ MC q ðϕ, γÞ is the meromorphic analogue of the class S q ðϕÞ introduced and studied by Mohd and Darus [19]. For γ = 0, the class ∑ MS q ðϕÞ was studied by Zayed et al. [20]. Definition 2. Let ∑ MK q ðϕ, ηÞ be the subclass of ∑ 1 consisting of all functions λðωÞ for which their exist δðωÞ ∈ ∑ MS ðαÞ and satisfy For η = 0 and δðωÞ = λðωÞ, the class ∑ MS q ðϕÞ was studied by Zayed et al. [20].

Main Results
In this section, we explore certain Fekete-Szego-related inequalities for the class ∑ MC q ðϕ, γÞ and ∑ MK q ðϕ, ηÞ.
Proof. Let λðωÞ ∈ ∑ MC q ðϕ, γÞ, then there exist analytic functions φðωÞ and kðωÞ, with jφðωÞj < 1, kð0Þ = 0, and kðωÞ < 1 such that Taking first and second derivative of (1), and use in the left hand side of above equation, we obtain 2 Journal of Function Spaces then implies which implies Comparing (13) and (16), we get Thus, Since φðωÞ is analytic and bounded in U * (see [25]), so we have By using this fact and the well-known inequality jk 1 j ≤ 1, we get Corollary 5. For φðωÞ = 1 and γ = 0 in Theorem 4, we obtain the result by Silverman et al. [3] (see Theorem 4).
For kðωÞ = ω and repeating steps of Theorem 4, we obtain the following corollary.
Proof. Let λðωÞ, δðωÞ ∈ ∑ MK q ðϕ, ηÞ, then there exist analytic functions φðωÞ and kðωÞ, with jφðωÞj < 1, kð0Þ = 0, and kðωÞ < 1 such that Taking first derivative of (1) and (5), and use in the left hand side of above equation, we obtain 3 Journal of Function Spaces then implies which implies Comparing (26) and (28), we get Thus, Since φðωÞ is analytic and bounded in U * (see [25]), we have By using this fact and the well-known inequality jk 1 j ≤ 1, we get Corollary 8. For φðωÞ = 1, δðωÞ = λðωÞ, and η = 0 in Theorem 7, we obtain the result by Silverman et al. [3] (see Theorem 7). For kðωÞ = ω and repeating steps of Theorem 7, we obtain the following corollary.

Meromorphic Functions Related with the Bessel Function
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 ν, α, and β are positive real numbers. The operator ζ ν,α,β is a meromorphic analogue introduced by Deniz [27] (see also Baricz et al. [28]) for analytic functions. In terms of convolution, ζ ν,α,β is given by The operator ζ ν,α,β 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.
Proof. Let λðωÞ ∈ ∑ MCðqÞ ν,α,β ðϕ, γÞ, then there exist analytic functions φðωÞ and kðωÞ, with jφðωÞj < 1, kð0Þ = 0, and kðωÞ < 1 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 φðωÞ is analytic and bounded in U * (see [25]), we have jc n j ≤ 1 − jc 0 j 2 ≤ 1, ðn > 0Þ: By using this fact and the well-known inequality jk 1 j ≤ 1, we get We have thus completed the proof of Theorem 12.

Journal of Function Spaces
For kðωÞ = ω and repeating steps of Theorem 12, we obtain the following corollary.