# Application of Quasisubordination to Certain Classes of Meromorphic Functions

Authors: Syed Ghoos Ali Shah; Saqib Hussain; Akhter Rasheed; Zahid Shareef; Maslina Darus
Journal: Journal of Function Spaces
 (2020)

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## 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.


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## Body

## 1. 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:
(1)λω=1ω+∑t=0∞atωt,which are analytic in the punctured open unit disc U∗=ω:ω∈Cand0<ω<1=U−0, as open unit disc U=U∗∪0.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 ∑MSα, we mean the subclass of ∑1 consisting of all meromorphic starlike functions of orderα. Analytically,
(2)λω∈∑MSα⇔Reωλ′ωλω<−α,0≤α<1;ω∈U∗.A closely related class of meromorphic convex functions of orderα is denoted by ∑MCα and defined as
(3)λω∈∑MCα⇔−ωλ′ω∈∑MSα.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 ∑MCα,β, of meromorphic close-to-convex functions of order α and type β if there existδω∈∑MSβ, and
(4)Reωλ′ωδω<−α.Letδω∈∑1 and having series representation of the form
(5)δω=1ω+∑t=0∞btωt.Then, the convolution ofλ and δ as denoted by λ∗δ is defined as
(6)λ∗δω=1ω+∑t=0∞atbtωt=δ∗λω,where λ is given by (1).A functionλ is subordinate to δ in U∗ and written as λω≺δω, if there exists a Schwarz function kω, which is holomorphic in U with k0=0, such that λω=δkω. Let ϕω be an analytic function with positive real part on U satisfies ϕ0=1 and ϕ′0>0 which maps U which is star shape with respect to ω=1, also symmetric with respect to the real axis. We denote ∑ϕ be the class of function λ∈∑1 for which −ωλ′ω/λω≺ϕω,ω∈U∗. The class ∑ϕ was introduced and studied by Silverman et al. [3] (see also [4]). The class ∑α is a special case of the class ∑ϕ when ϕω=1+1−2αω/1−ω0≤α<1.Robertson [5] gave the idea of quasisubordination. For any two functions λω and δω, holomorphic in U, the function λω is said to be quasi-subordinate to the function δω written as λω≺qδω, if there exists two holomorphic functions λω and φω, with φω≤1, λω/φω is holomorphic in U∗ and such that λω=φωδkω. In particular, if φω=1, then quasisubordination reduces to subordination. Furthermore, if kω=1, the quasisubordination becomes the majorization, (see [6]), which implies
(7)λω≺qδω⇒λω=φωδω⇒λω<<δωω∈U∗.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 assume0≤γ<1, γ≠1/2, 0≤η<1, ω∈U∗,λ,δ∈∑1, and ϕω be an analytic function with positive real part on U that satisfies ϕ0=1 and ϕ′0>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∑qMCϕ,γ be the class of functions λω∈∑1 and satisfy
(8)−1−γωλ′ωλω+γ1+ωλ′′ωλ′ω−1≺qϕω−1,ω∈U∗.The abovementioned class∑qMCϕ,γ is the meromorphic analogue of the class Sqϕ introduced and studied by Mohd and Darus [19]. For γ=0, the class ∑qMSϕ was studied by Zayed et al. [20].Definition 2.
Let∑qMKϕ,η be the subclass of ∑1 consisting of all functions λω for which their exist δω∈∑MSα and satisfy
(9)−ωλ′ω1−ηδω+ηωδ′ω−1≺qϕω−1,ω∈U∗.Forη=0 and δω=λω, the class ∑qMSϕ was studied by Zayed et al. [20].In this paper, we obtain the Fekete-Szego inequality for meromorphic functions belonging to above defined classes. Letk∈Ω denote the class of functions of the form kω=k1ω+k2ω2+k3ω3+⋯, satisfying kω<1, for ω∈U∗. For more details, see [21–23]. To prove our main results, we need the following lemma.Lemma 3 [24].
Ifk∈Ω, then for any complex number u, k2−uk12≤max1;u, the result is sharp for the functions given by kω=ωorkω=ω2.


## 2. Main Results

In this section, we explore certain Fekete-Szego-related inequalities for the class∑qMCϕ,γ and ∑qMKϕ,η.Theorem 4.
Letϕω=1+B1ω+B2ω2+⋯.,B1>0, and φω=c0+c1ω+c2ω2+⋯ if λω given by (1) be in the class ∑qMCϕ,γ, and μ is a complex number, then
(10)a1−μa02≤B121−2γ1+max1,B2B1+B111−γ−2μ1−2γ1−γ2.The inequalities are sharp forkω=ω or kω=ω2.Proof.
Letλω∈∑qMCϕ,γ, then there exist analytic functions φω and kω, with φω<1, k0=0, and kω<1 such that
(11)−1−γωλ′ωλω+γ1+ωλ′′ωλ′ω−1=φωϕkω−1,ω∈U∗.
Taking first and second derivative of (1), and use in the left hand side of above equation, we obtain
(12)−1−γωλ′ωλω+γ1+ωλ′′ωλ′ω=1−γ1−a0ω+a02−2a1ω2+γ1+2a1ω2+⋯,then implies
(13)−1−γωλ′ωλω+γ1+ωλ"ωλ′ω−1=−a0ω1−γ+a02−2a1+γ−a02+4a1ω2+⋯,(14)ϕkω=ϕk1ω+k2ω2+k3ω3+⋯=1+k1B1ω+k12B2+k2B1ω2+k3B1+2k1k2B2+k13B3ω3+⋯,then
(15)φωϕkω−1=c0+c1ω+c2ω2+⋯1+k1B1ω+k12B2+k2B1ω2+k3B1+2k1k2B2+k13B3ω3+⋯,which implies
(16)φωϕkω−1=c0k1B1ω+c0k12B2+c0k2B1+c1k1B1ω2+⋯.
Comparing (13) and (16), we get
(17)a0=−c0k1B11−γ,a1=−B1c021−2γk2+k1c1c0+k12B2B1−B1c01−γ.
Thus,(18)a1−μa02=−B1c021−2γk2+k1c1c0+k12B2B1−B1c01−γ+2μc0B11−2γ1−γ2,a1−μa02≤−B1c021−2γk1c1c0+k2+k12B2B1−B1c01−γ+2μc0B11−2γ1−γ2.
Sinceφω is analytic and bounded in U∗ (see [25]), so we have
(19)cn≤1−c02≤1,n>0.
By using this fact and the well-known inequalityk1≤1, we get
(20)a1−μa02≤B121−2γ1+k2+k12B2B1−B1c01−γ+2μc0B11−2γ1−γ2.Corollary 5.
Forφω=1 and γ=0 in Theorem 4, we obtain the result by Silverman et al. [3] (see Theorem 4).Forkω=ω and repeating steps of Theorem 4, we obtain the following corollary.Corollary 6.
Letλω∈∑1 satisfies
(21)−1−γωλ′ωλω+γ1+ωλ′′ωλ′ω−1<<ϕω−1,ω∈U∗,then for any complex numberμ,
(22)a1−μa02≤B121−2γ1+B2B1+B111−γ−2μ1−2γ1−γ2.Theorem 7.
Letω=1+B1ω+B2ω2+⋯,B1>0, and φω=c0+c1ω+c2ω2+⋯, if λω, δω, given by (1) and (5) be in the class ∑qMKϕ,η, and μ is a complex number, then
(23)b1+a12−a1η−μb02≤B121+max1,B2B1+B11−1−η2−2μ1−η2−1−4η2.The inequalities is sharp forkω=ω or kω=ω2.Proof.
Letλω, δω∈∑qMKϕ,η, then there exist analytic functions φω and kω, with φω<1, k0=0, and kω<1 such that
(24)−ωλ′ω1−ηδω+ηωδ′ω−1=φωϕkω−1,ω∈U∗.
Taking first derivative of (1) and (5), and use in the left hand side of above equation, we obtain
(25)−ωλ′ω1−ηδω+ηωδ′ω=1+2η1+2η+b01−η−1−4ηω+b021−η2−b1+2a1η−a1ω2+⋯,
then implies(26)−ωλ′ω1−ηδω+ηωδ′ω−1=2η1+2η+b01−η−1−4ηω+b021−η2−b1+2a1η−a1ω2+⋯,(27)ϕkω=1+k1B1ω+k12B2+k2B1ω2+k3B1+2k1k2B2+k13B3ω3+⋯,
which implies(28)φωϕkω−1=c0k1B1ω+c0k12B2+c0k2B1+c1k1B1ω2+⋯.
Comparing (26) and (28), we get
(29)b0=c0k1B11−η−1−4η,b1+a12−a1η=−B1c02k2+k1c1c0+k12B2B1−B1c0−1−4η2.
Thus,(30)b1+a12−a1η−μb02=−B1c02k2+k1c1c0+k12B2B1−B1c0−1−4η2+2μc0B11−η2−1−4η2.
Sinceφω is analytic and bounded in U∗ (see [25]), we have
(31)cn≤1−c02≤1,n>0.
By using this fact and the well-known inequalityk1≤1, we get
(32)b1+a12−a1η−μb02≤B121+k2+k12B2B1−B1c0−1−4η2+2μc0B11−η2−1−4η2.Corollary 8.
Forφω=1, δω=λω, and η=0 in Theorem 7, we obtain the result by Silverman et al. [3] (see Theorem 7). For kω=ω and repeating steps of Theorem 7, we obtain the following corollary.Corollary 9.
Letλω and δω∈∑1 satisfy
(33)−ωλ′ω1−ηδω+ηωδ′ω−1<<ϕω−1,ω∈U∗,then for any complex number μ,
(34)b1+a12−a1η−μb02≤B121+B2B1+B11−1−4η2−2μ1−η2−1−4η2.


## 3. Meromorphic Functions Related with the Bessel Function

Let us consider the second order linear homogenous differential equation (see, Baricz [26])
(35)ω2k′′ω+αωk′ω+βω2−ν2+1−αkω=0.The function(36)kν,α,βω=∑t=0∞−βtΓt+1Γt+ν+1+α+1/2ω22t+νis known as generalized Bessel’s function of first kind and is the solution of differential equation given in (35). If we denote
(37)ζν,α,βλω=2νΓν+α+1/2ων/2+1kν,α,βω1/2,=1ω+∑t=0∞−βt+1Γν+α+1/24t+1Γt+2Γt+ν+1+α+1/2ωt,where ν, α, and β are positive real numbers. The operator ζν,α,β is a meromorphic analogue introduced by Deniz [27] (see also Baricz et al. [28]) for analytic functions. In terms of convolution, ζν,α,β is given by
(38)ζν,α,βλω=kν,α,βω∗λω=1ω+∑t=0∞−βt+1Γν+α+1/24t+1Γt+2Γt+ν+1+α+1/2atωt.The operatorζν,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.Definition 10.
A functionλω∈∑1 given by (1) is said to belong to the class ∑ν,α,βMCqϕ,γ if
(39)−1−γωζν,α,βλ′ωζν,α,βλω+γ1+ωζν,α,βλ′′ωζν,α,βλ′ω−1≺qϕω−1.Forγ=0, the class ∑ν,α,βMSqϕ was studied by Zayed et al. [20].Definition 11.
Let∑ν,α,βMKqϕ,η be the subclasses of ∑1consisting of all functions λω∈∑1 for which there exist δω∈∑MSα, and
(40)−ωζν,α,βλ′ω1−ηζν,α,βδω+ηωζν,α,βδ′ω−1≺qϕω−1ω∈U∗.Forη=0 and δω=λω, the class ∑ν,α,βMSqϕ was studied by Zayed et al. [20].Theorem 12.
Letω=1+B1ω+B2ω2+⋯.,B1>0, and φω=c0+c1ω+c2ω2+⋯., if λω given by (1) be in the class ∑ν,α,βMCqϕ,γ, and μ is a complex number, then
(41)a1−μa02≤42ν+α+1/2ν+1+α+1/2B1β21−2γ×1+max1,B2B1+B11−μν+α+1/21−2γν+1+α+1/21−γ.Proof.
Letλω∈∑ν,α,βMCqϕ,γ, then there exist analytic functions φω and kω, with φω<1, k0=0, and kω<1 such that
(42)−1−γωζν,α,βλ′ωζν,α,βλω+γ1+ωζν,α,βλ′′ωζν,α,βλ′ω−1=φωϕkω−1.
Taking first and second derivative of (38), in use of the left side of the above equation, we obtain
(43)−1−γωζν,α,βλ′ωζν,α,βλω+γ1+ωζν,α,βλ′′ωζν,α,βλ′ω=1−γ1+βa0ω4ν+α+1/2−2β2a1ω242×2ν+1+α+1/2ν+α+1/2+β2a02ω242ν+α+1/22+⋯+γ1+2β2a1ω242×2ν+1+α+1/2ν+α+1/2+⋯,thus implies
(44)−1−γωζν,α,βλ′ωζν,α,βλω+γ1+ωζν,α,βλ′′ωζν,α,βλ′ω−1=βa04ν+α+1/21−γω+β2a0242ν+α+1/221−γ−2β2a142×2ν+1+α+1/2ν+α+1/21−2γω2+⋯,(45)ϕkω=1+k1B1ω+k12B2+k2B1ω2+k3B1+2k1k2B2+k13B3ω3+⋯,which implies
(46)φωϕkω−1=c0k1B1ω+c0k12B2+c0k2B1+c1k1B1ω2+⋯.
Comparing (44) and (46), we get
(47)a0=4ν+α+1/2c0k1B1β1−γ,a1=−42ν+1+α+1/2ν+α+1/2B1β21−2γc0k2+c1k1+k12c0B2B1−c0B11−γ.
Thus,(48)a1−μa02=−42ν+1+α+1/2ν+α+1/2B1β21−2γ×c0k2+k12c1k1+c0B2B1−B1c01−γ−ν+α+1/21−2γc0ν+1+α+1/21−γ,a1−μa02≤−42ν+1+α+1/2ν+α+1/2B1β21−2γ×c0k2+k12c1k1+c0B2B1−B1c01−γ−ν+α+1/21−2γc0ν+1+α+1/21−γ.
Sinceφω is analytic and bounded in U∗ (see [25]), we have cn≤1−c02≤1, n>0.
By using this fact and the well-known inequalityk1≤1, we get
(49)a1−μa02≤42ν+α+1/2ν+1+α+1/2B1β21−2γ×1+max1,B2B1+B11−μν+α+1/21−2γν+1+α+1/21−γ.We have thus completed the proof of Theorem12.Forkω=ω and repeating steps of Theorem 12, we obtain the following corollary.Corollary 13.
Letλω∈∑1 satisfies
(50)−1−γωζν,α,βλ′ωζν,α,βλω+γ1+ωζν,α,βλ′′ωζν,α,βλ′ω−1≪ϕω−1ω∈U∗.Then, for any complex numberμ,
(51)a1−μa02≤42ν+α+1/2ν+1+α+1/2B1β21−2γ1+B2B1+B111−γ−μν+α+1/21−2γν+1+α+1/2.Theorem 14.
Letϕω=1+B1ω+B2ω2+⋯.,B1>0 , and φω=c0+c1ω+c2ω2+⋯., if λω and δω given by (1) and (5) be in the class ∑ν,α,βMKqϕ,η and μ is a complex number, then
(52)b1+a12−b1η−μb02≤42ν+α+1/2ν+1+α+1/2B1β2×1+max1,B2B1+B11−μν+α+1/2ν+1+α+1/21−η2.Proof.
Letλω and δω∈∑ν,α,βMKqϕ,η then there exist analytic functions φω and kω, with φω<1, k0=0, and kω<1 such that
(53)−ωζν,α,βλ′ω1−ηζν,α,βδω+ηωζν,α,βδ′ω−1=φωϕkω−1.
Taking first derivative of (5) and (38) in use of the left side of above equation, we have
(54)−ωζν,α,βλ′ω1−ηζν,α,βδω+ηωζν,α,βδ′ω=1+βb01−η4ν+α+1/2ω+β2b021−η242ν+α+1/22−β2b11−2η42×2ν+1+α+1/2ν+α+1/2−β2a142×2ν+1+α+1/2ν+α+1/2ω2+⋯,and this implies
(55)−ωζν,α,βλ′ω1−ηζν,α,βδω+ηωζν,α,βδ′ω−1=βb01−η4ν+α+1/2ω+β2b021−η242ν+α+1/22−β2b11−2η42×2ν+1+α+1/2ν+α+1/2−β2a142×2ν+1+α+1/2ν+α+1/2ω2+⋯,(56)ϕkω=1+k1B1ω+k12B2+k2B1ω2+k3B1+2k1k2B2+k13B3ω3+⋯,which implies
(57)φωϕkω−1=c0k1B1ω+c0k12B2+c0k2B1+c1k1B1ω2+⋯.
Comparing (55) and (57), we get
(58)b0=4ν+α+1/2c0k1B1β1−ηω,b1+a12−b1η=−42ν+α+1/2ν+1+α+1/2B1β2c0k2+c1k1+k12c0B2B1−c02B1.
Thus,(59)b1+a12−b1η−μb02=−42ν+α+1/2ν+1+α+1/2B1β2×c0k2+c1k1+k12c0B2B1−c02B1+μν+α+1/2c02B1ν+1+α+1/21−η2,b1+a12−b1η−μb02≤−42ν+α+1/2ν+1+α+1/2B1β2×c0k2+k12c0k2+c0B2B1−B1c02−μν+α+1/2c02ν+1+α+1/21−η2.
Sinceφω is analytic and bounded in U∗ (see [25]), we have
(60)cn≤1−c02≤1,n>0.
By using this fact and the well-known inequality,k1≤1, we get
(61)b1+a12−b1η−μb02≤42ν+α+1/2ν+1+α+1/2B1β2×1+max1,B2B1+B11−μν+α+1/2ν+1+α+1/21−η2.We have thus completed the proof of Theorem14.Forkω=ω and repeating steps of Theorem 14, we obtain the following corollary.Corollary 15.
Letλω and δω satisfy
(62)−ωζν,α,βλ′ω1−ηζν,α,βδω+ηωζν,α,βδ′′ω−1<<ϕω−1,ω∈U∗,then for any complex number μ,
(63)b1+a12−b1η−μb02≤42ν+α+1/2ν+1+α+1/2B1β2×1+B2B1+B11−μν+α+1/2ν+1+α+1/21−η2.


## 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.



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