# Convolution Results and Fekete–Szegö Inequalities for Certain Classes of Symmetricq-Starlike and Symmetric q-Convex Functions

Authors: Tamer M. Seoudy
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, by using the concept of the symmetricq-difference operator, we introduce certain classes of symmetric q-starlike and symmetric q-convex functions. Convolution results, coefficient estimates, and Fekete–Szegö inequalities for the analytic functions belonging to these classes are obtained.


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## Body

## 1. Introduction

Let us represent the class of analytic (or holomorphic or regular) functions inΔ=ξ∈ℂ:ξ<1 by ℋΔ and suppose that A is the subclass of ℋΔ defined as follows:(1)A=g∈ℋΔ:gξ=ξ+∑j=2∞bjξj.Further, letSα and Cα denote the subclasses of A that consist, respectively, of starlike of order α and convex of order α0≤α<1 in Δ (see [1]). For two given regular functions h1,h2∈ℋΔ, we say that h1ξ is subordinate to h2ξ or h2ξ is superordinate to h1ξ, written symbolically as h1ξ≺h2ξ if there exists a Schwarz function ω, which (by definition) is regular in Δ with ω0=0 and ωξ<1 for all ξ∈Δ, such that(2)h1ξ=h2ωξ,ξ∈Δ.Moreover, ifh1ξ is univalent function in Δ, then the following equivalence holds true (see [2, 3]):(3)h1ξ≺h2ξ⇔h10=h20,h1Δ⊂h2Δ.For functionsg given by (1) and h given by(4)hξ=ξ+∑j=2∞cjξj,the convolution or Hadamard product of g and h is defined by(5)g∗hξ=ξ+∑j=2∞bjcjξj=h∗gξ.LetSL,M and CL,M denote the subclasses of A for −1≤M<L≤1 which are defined by (see [4–9])(6)SL,M=gξ∈A:ξg′ξgξ≺1+Lξ1+Mξ,CL,M=gξ∈A:ξg′ξ′g′ξ≺1+Lξ1+Mξ.We note thatS1−2α,−1=Sα and C1−2α,−1=Cα0≤α<1.For0<q<1, the symmetric q-difference of a function g is defined as follows (see [10]):(7)D˜qgξ=gqξ−gq−1ξq−q−1ξ,ξ≠0,and D˜qg0=g′0 provided that gξ is differentiable at 0. From (1) and (7), we deduce that(8)D˜qgξ=1+∑j=2∞j˜qbjξj−1,where j˜q is the q-number given by(9)j˜q=qj−q−jq−q−1.Furthermore, ifgξ and hξ are the two functions, then(10)D˜qγ1gξ±γ2hξ=γ1D˜qgξ±γ2D˜qhξ,where γ1andγ2 are constants.(11)D˜qgξhξ=hqξD˜qgξ+gq−1ξD˜qhξ,D˜qgξhξ=hqξD˜qgξ−gqξD˜qhξhqξhq−1ξ.Making use of the symmetricq-difference D˜qgξ given by (7), we introduce the subclasses S˜qL,M and C˜qL,M of A for 0<q<1 and −1≤M<L≤1 as follows:(12)S˜qL,M=gξ∈A:ξD˜qgξgξ≺1+Lξ1+Mξ,(13)C˜qL,M=gξ∈A:D˜qξD˜qgξD˜qgξ≺1+Lξ1+Mξ.From (12) and (13), we have(14)gξ∈C˜qL,M⇔ξD˜qgξ∈S˜qL,M.We also note that(i)
limq⟶1−S˜qL,M=SL,M and limq⟶1−C˜qL,M=CL,M (see [6, 7, 9, 11, 12]);(ii)
S˜q1−2α,−1=S˜qα0≤α<1,(15)S˜qα=gξ∈A:ℜξD˜qgξgξ>α,ξ∈Δ,andlimq⟶1−S˜qα=Sα (see [1, 13, 14]).(iii)
C˜q1−2α,−1=C˜qα0≤α<1,(16)C˜qα=gξ∈A:ℜD˜qξD˜qgξD˜qgξ>α,ξ∈Δ,andlimq⟶1−C˜qα=Cα (see [1, 8, 15]).(iv)
S˜q1−2αβ,−β=S˜qα,β0≤α<1,0<β≤1,(17)S˜qα,β=gξ∈A:ξD˜qgξ/gξ−1ξD˜qgξ/gξ+1−2α<β,ξ∈Δ,andlimq⟶1−S˜qα,β=Sα,β (see [16]).(v)
C˜q1−2αβ,−β=C˜qα,β0≤α<1,0<β≤1,(18)C˜qα,β=gξ∈A:D˜qξD˜qgξ/D˜qgξ−1D˜qξD˜qgξ/D˜qgξ+1−2α<β,ξ∈Δ,and limq⟶1−C˜qα,β=Cα,β (see [16]).In the present article, our aim is to investigate convolution properties, coefficient estimates, and Fekete–Szegö inequalities for the classesS˜qL,M and C˜qL,M. The motivation of this article is to generalize and improve previously known results.


## 2. Convolution Results and Coefficient Estimates

Unless otherwise mentioned, we assume throughout this investigation that(19)0≤θ<2π,−1≤M<L≤1,0<q<1,ξ∈Δ.Theorem 1.
Ifgξ∈A has the series form (1), then gξ∈S˜qL,M if and only if(20)1ξgξ∗ξ−1+Rq−1+q−1ξ2+Rξ31−qξ1−ξ1−q−1ξ≠0,where R is given by(21)R=Rθ,L,M=e−iθ+LL−M.Proof.
It is easy to check that(22)gξ∗ξ1−ξ=gξ,gξ∗ξ1−qξ1−q−1ξ=ξD˜qgξ.
In order to prove that equation (20) holds, we will write (12) by using the definition of the subordination as follows:(23)ξD˜qgξgξ=1+Lwξ1+Mwξ,where wξ is Schwarz function; hence, we have(24)ξD˜qgξgξ≠1+Leiθ1+Meiθ,which is equivalent to(25)1ξ1+MeiθξD˜qgξ−1+Leiθgξ≠0.
By using (22) in (25), we obtain(26)1ξ1+Meiθgξ∗ξ1−qξ1−q−1ξ−1+Leiθgξ∗ξ1−ξ=M−Leiθξgξ∗ξ−1+e−iθ+L/L−Mq−1+q−1ξ2+e−iθ+L/L−Mξ31−qξ1−ξ1−q−1ξ≠0,which proves the necessary condition (20) for Theorem 1.
Reversely, suppose thatgξ∈A satisfies condition (20). Since it was shown in the first part of the proof that assumption (20) is equivalent to (25), we obtain that(27)ξD˜qgξgξ≠1+Leiθ1+Meiθ,0≤θ<2π.
If we denote(28)φξ=ξD˜qgξgξ,ψξ=1+Lξ1+Mξ,relation (27) means that(29)φΔ∩ψ∂Δ=∅.
Thus, the simply connected domainφΔ is included in a connected component of ℂ\ψ∂Δ. Therefore, using the fact that φ0=ψ0 and the univalence of the function ψξ, it follows that φξ≺ψξ, which implies that gξ∈S˜qL,M. This completes the proof of Theorem 1.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 1, we get the following convolution result for S˜qα.Corollary 1.
Ifgξ∈A has the series form (1), then gξ∈S˜qα if and only if(30)1ξgξ∗ξ−1+Tq−1+q−1ξ2+Tξ31−qξ1−ξ1−q−1ξ≠0,where T is given by(31)T=Tθ,α=e−iθ+1−2α21−α,0≤α<1.Theorem 2.
Ifgξ∈A has the series form (1), then gξ∈C˜qL,M if and only if(32)1ξgξ∗ξ−Rq+q−1+1q−1+q−1ξ2+Rq+2+q−1−1q−1+q−1ξ3−2R−1ξ41−q2ξ1−qξ1−ξ1−q−1ξ1−q−2ξ≠0,where R is given by (21).Proof.
From relation (14), we have(33)gξ∈C˜qL,M⇔ξD˜qgξ∈S˜qL,M.
Then, according to Theorem1, we obtain(34)gξ∈C˜qL,M⇔1ξξD˜qgξ∗ϕξ≠0,where ϕξ is given by(35)ϕξ=ξ−1+Rq−1+q−1ξ2+Rξ31−qξ1−ξ1−q−1ξ,and we note that(36)ξD˜qϕξ=ξ−Rq+q−1+1q−1+q−1ξ2+Rq+2+q−1−1q−1+q−1ξ3−2R−1ξ41−q2ξ1−qξ1−ξ1−q−1ξ1−q−2ξ.
Using relation (36) and the following identity:(37)ξD˜qgξ∗ϕξ=gξ∗ξD˜qϕξ,g,ϕ∈A,it is easy to check that (34) is equivalent to (32). Thus, the proof of Theorem 2 is completed.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 2, we obtain the following result for C˜qα.Corollary 2.
Ifgξ∈A has the series form (1), then gξ∈C˜qα0≤α<1 if and only if(38)1ξgξ∗ξ−Tq+q−1+1q−1+q−1ξ2+Tq+2+q−1−1q−1+q−1ξ3−2T−1ξ41−q2ξ1−qξ1−ξ1−q−1ξ1−q−2ξ≠0,where T is given by (31).Theorem 3.
Ifgξ∈A has the series form (1), then gξ∈S˜qL,M if and only if(39)1−∑j=2∞j˜q−1e−iθ+M−L+ML−Mbjξj−1≠0.Proof.
From Theorem1, we find that gξ∈S˜qL,M if and only if(40)1ξgξ∗ξ−1+Rq−1+q−1ξ2+Rξ31−qξ1−ξ1−q−1ξ≠0,for all R given by (21). The left hand side of (38) can be written as(41)1ξgξ∗Rξ1−ξ−R−1ξ1−qξ1−q−1ξ=1ξRgξ−R−1ξD˜qgξ=1−∑j=2∞j˜qR−1−Rbjξj−1=1−∑j=2∞j˜q−1e−iθ+M−L+ML−Mbjξj−1.
Thus, the proof of Theorem3 is completed.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 3, we obtain the following result for S˜qα.Corollary 3.
Ifgξ∈A has the series form (1), then gξ∈S˜qα if and only if(42)1−∑j=2∞j˜q−1e−iθ−1−21−α21−αbjξj−1≠0.Theorem 4.
Ifgξ∈A has the series form (1), then gξ∈C˜qL,M if and only if(43)1−∑j=2∞j˜qj˜q−1e−iθ+M−L+ML−Mbjξj−1≠0ξ∈Δ.Proof.
From Theorem1, we find that gξ∈C˜qL,M if and only if(44)1ξgξ∗ξ−Rq+q−1+1q−1+q−1ξ2+Rq+2+q−1−1q−1+q−1ξ3−2R−1ξ41−q2ξ1−qξ1−ξ1−q−1ξ1−q−2ξ≠0,for all R given by (21). The left hand side of (44) can be written as(45)1ξgξ∗Rξ1−qξ1−q−1ξ−R−1ξ+ξ21−q2ξ1−ξ1−q−2ξ1−ξ=1ξRξD˜qgξ−R−1ξD˜qξD˜qgξ=1−∑j=2∞j˜qj˜qR−1−Rbjξj−1=1−∑j=2∞j˜qj˜q−1e−iθ+M−L+ML−Mbjξj−1,and this proves Theorem 4.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 4, we obtain the following convolution result for C˜qα.Corollary 4.
Ifgξ∈A has the series form (1), then gξ∈S˜qα if and only if(46)1−∑j=2∞j˜qj˜q−1e−iθ−1−21−α21−αbjξj−1≠0.Theorem 5.
If the functiongξ∈A has the series form (1) satisfying the inequality(47)∑j=2∞j˜q−11−M+L−Mbj≤L−M,then gξ∈S˜qL,M.Proof.
Hence,(48)1−∑j=2∞j˜q−1e−iθ+M+M−LL−Mbjξj−1≥1−∑j=2∞j˜q−1e−iθ+M+M−LL−Mbjξj−1≥1−∑j=2∞j˜q−11−M+L−ML−Mbj>0.
Thus, inequality (47) holds, and our result follows from Theorem 3.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 5, we obtain the following result for S˜qα.Corollary 5.
If the functiongξ∈A has the series form (1) satisfying the inequality(49)∑j=2∞j˜q−αbj≤1−α,then gξ∈S˜qα.By using arguments and analysis to those in the proof of Theorem5, we can derive the following theorem.Theorem 6.
If the functiongξ∈A has the series form (1) satisfying the inequality(50)∑j=2∞j˜qj˜q−11−M+L−Mbj≤L−M,then gξ∈C˜qL,M.PuttingL=1−2α0≤α<1 and M=−1 in Theorem 6, we obtain the following result for C˜qα.Corollary 6.
If the functiongξ∈A has the series form (1) satisfying the inequality(51)∑j=2∞j˜qj˜q−αbj≤1−α,then gξ∈C˜qα.


## 3. Fekete–Szegö Inequalities

In this section, we obtain the Fekete–Szegö inequalities for the classesS˜qL,M and C˜qL,M. In order to establish our results, we need the following lemmas.Lemma 1 (see [17]).
Ifpξ=1+δ1ξ+δ2ξ2+⋯ is a function with positive real part in Δ and v is a complex number, then(52)δ2−vδ12≤2max1;2v−1.The result is sharp for the functions given by(53)pξ=1+ξ21−ξ2,pξ=1+ξ1−ξ.Lemma 2 (see [17]).
Ifpξ=1+δ1ξ+δ2ξ2+⋯ is an analytic function with a positive real part in Δ, then(54)δ2−νδ12≤−4ν+2,ifν≤0,2,if0≤ν≤1,4ν−2,ifν≥1.Whenυ<0 or ν>1, the equality holds if and only if pξ is 1+ξ/1−ξ or one of its rotations. If 0<ν<1, then the equality holds if and only if pξ is 1+ξ2/1−ξ2 or one of its rotations. If ν=0, the equality holds if and only if(55)pξ=1+λ21+ξ1−ξ+1−λ21−ξ1+ξ,0≤λ≤1,or one of its rotations. If ν=1, the equality holds if and only if pξ is the reciprocal of one of the functions such that equality holds in the case of ν=0.Also, the above upper bound is sharp, and it can be improved as follows when0<ν<1:(56)δ2−νδ12+νδ12≤2,0≤ν≤12,δ2−νδ12+1−νδ12≤2,12≤ν≤1.Theorem 7.
Ifgξ defined by (1) belongs to the class S˜qL,M, then(57)b3−μb22≤L−M3˜q−1max1;M−L−M2˜q−11−3˜q−12˜q−1μ.Proof.
Ifgξ∈S˜qL,M, then there is a Schwarz function w in Δ such that(58)ξD˜qgξgξ=1+Lwξ1+Mwξ.
Define the functionpξ by(59)pξ=1+wξ1−wξ=1+δ1ξ+δ2ξ2+⋯.
Sincewξ is a Schwarz function, we see that ℜpξ>0 and p0=1. Now, by substituting (59) in (58), we have(60)ξD˜qgξgξ=1+L−M2δ1ξ+L−M2δ2−1+M2δ12ξ2+⋯.
From the above equation, we obtain(61)2˜q−1b2=L−M2δ1,3˜q−1b3−2˜q−1b22=L−M2δ2−1+M2δ12,or, equivalently,(62)b2=L−M22˜q−1δ1,b3=L−M23˜q−1δ2−121+M−L−M2˜q−1δ12.
Therefore, we have(63)b3−μb22=L−M23˜q−1δ2−vδ12,where(64)ν=121+M−L−M2˜q−11−3˜q−12˜q−1μ.
Our result now follows from Lemma1. This completes the proof of Theorem 1.
Similarly, we can prove the following theorem for the classC˜qL,M.Theorem 8.
Ifgξ given by (1) belongs to the class C˜qL,M, then(65)b3−μb22≤L−M3˜q3˜q−1max1;M−L−M2˜q−11−3˜q3˜q−12˜q22˜q−1μ.The result is sharp.PuttingL=1−2α0≤α<1 and M=−1 in Theorems 7 and 8, we obtain the following corollaries.Corollary 7.
Ifgξ given by (1) belongs to the class S˜qα0≤α<1, then(66)b3−μb22≤21−α3˜q−1max1;1+21−α2˜q−11−3˜q−12˜q−1μ.Corollary 8.
Ifgξ given by (1) belongs to the class C˜qα0≤α<1, then(67)b3−μb22≤21−α3˜q3˜q−1max1;1+21−α2˜q−11−3˜q3˜q−12˜q22˜q−1μ.The results are sharp.Theorem 9.
Let(68)σ1=2˜q−1L−M−2˜q−121+M3˜q−1L−M,σ2=2˜q−1L−M+2˜q−121−M3˜q−1L−M,σ3=2˜q−1L−M−2˜q−12M3˜q−1L−M.Ifg given by (1) belongs to the class S˜qL,M, then(69)b3−μb22≤−L−M3˜q−1M−L−M2˜q−11−3˜q−12˜q−1μ,μ≤σ1,L−M3˜q−1,σ1≤μ≤σ2,L−M3˜q−1M−L−M2˜q−11−3˜q−12˜q−1μ,μ≥σ2.Further, ifσ1≤μ≤σ3, then(70)b3−μb22+2˜q−123˜q−11+ML−M−12˜q−11−3˜q−12˜q−1μb22≤L−M3˜q−1.Ifσ3≤μ≤σ2, then(71)b3−μb22+2˜q−123˜q−11−ML−M+12˜q−11−3˜q−12˜q−1μb22≤L−M3˜q−1.Proof.
Applying Lemma2 to (63) and (64), we can obtain our results asserted by Theorem 9.
PuttingL=1−2α0≤α<1 and M=−1 in Theorem 9, we obtain the following result.Corollary 9.
Let(72)σ4=2˜q−13˜q−1,σ5=2˜q−11−α+2˜q−123˜q−11−α,σ6=22˜q−11−α+2˜q−1223˜q−11−α.Ifg given by (1) belongs to the class S˜qα0≤α<1, then(73)b3−μb22≤21−α3˜q−11+21−α2˜q−11−3˜q−12˜q−1μ,μ≤σ4,21−α3˜q−1,σ4≤μ≤σ5,−21−α3˜q−11+21−α2˜q−11−3˜q−12˜q−1μ,μ≥σ5.Further, ifσ4≤μ≤σ6, then(74)b3−μb22−2˜q−13˜q−11−3˜q−12˜q−1μb22≤21−α3˜q−1.Ifσ6≤μ≤σ5, then(75)b3−μb22+2˜q−123˜q−111−α+12˜q−11−3˜q−12˜q−1μb22≤21−α3˜q−1.Similarly, we can obtain the following theorem.Theorem 10.
Let(76)χ1=2˜q22˜q−1L−M−2˜q−11+M3˜q−1L−M,χ2=2˜q22˜q−1L−M+2˜q−11−M3˜q−1L−M,χ3=2˜q22˜q−1L−M−2˜q−1M3˜q−1L−M.Ifg given by (1) belongs to the class C˜qL,M, then(77)b3−μb22=−L−M3˜q3˜q−1M−L−M2˜q−11−3˜q3˜q−12˜q22˜q−1μ,μ≤χ1,L−M3˜q3˜q−1,χ1≤μ≤χ2,L−M3˜q3˜q−1M−L−M2˜q−11−3˜q3˜q−12˜q22˜q−1μ,μ≥χ2.Further, ifχ1≤μ≤χ3, then(78)b3−μb22+2˜q22˜q−123˜q3˜q−11+ML−M−12˜q−11−3˜q3˜q−12˜q22˜q−1μb22≤L−M3˜q3˜q−1.Ifχ3≤μ≤χ2, then(79)b3−μb22+2˜q22˜q−123˜q3˜q−11−ML−M+12˜q−11−3˜q3˜q−12˜q22˜q−1μb22≤L−M3˜q3˜q−1.PuttingL=1−2α0≤α<1 and M=−1 in Theorem 10, we obtain the following result.Corollary 10.
Let(80)χ4=2˜q22˜q−13˜q−1,χ5=2˜q22˜q−11−α+2˜q−13˜q−11−α,χ6=2˜q22˜q−121−α+2˜q−123˜q−11−α.Ifg given by (1) belongs to the class C˜qα0≤α<1, then(81)b3−μb22≤21−α3˜q3˜q−11+21−α2˜q−11−3˜q3˜q−12˜q22˜q−1μ,μ≤χ4,21−α3˜q3˜q−1,χ4≤μ≤χ5,−21−α3˜q3˜q−11+21−α2˜q−11−3˜q3˜q−12˜q22˜q−1μ,μ≥χ5.Further, ifχ4≤μ≤χ6, then(82)b3−μb22−2˜q22˜q−13˜q3˜q−11−3˜q3˜q−12˜q22˜q−1μb22≤21−α3˜q3˜q−1.Ifχ6≤μ≤χ5, then(83)b3−μb22+2˜q22˜q−123˜q3˜q−111−α+12˜q−11−3˜q3˜q−12˜q22˜q−1μb22≤21−α3˜q3˜q−1.


## 4. Conclusion

In this present investigation, we have introduced two classesS˜qL,M and C˜qL,M of analytic functions by using the symmetric q-difference operator linked to an open unit disc Δ=ξ∈ℂ:ξ<1. We also studied convolution results, coefficient estimates, and Fekete–Szegö inequalities for the newly defined classes. We note that our results naturally include several results that are known for those subclasses, which are listed in the introduction section.



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