# On Fekete-Szegö Problems for Certain Subclasses Defined byq-Derivative

Authors: Huda Aldweby; Maslina Darus
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We derive the Fekete-Szegö theorem for new subclasses of analytic functions which areq-analogue of well-known classes introduced before.


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## Body

## 1. Introduction

Denote byA the class of all analytic functions of the form (1)fz=z+∑k=2∞akzkin the open unit disk U={z∈C:|z|<1}.For two analytic functionsf and g in U, the subordination between them is written as f≺g. Frankly, the function f(z) is subordinate to g(z) if there is a Schwarz function w with w(0)=0, |w(z)|<1, for all z∈U, such that f(z)=g(w(z)) for all z∈U. Note that, if g is univalent, then f≺g if and only if f(0)=g(0) and f(U)⊆g(U).In [1, 2], Jackson defined the q-derivative operator Dq of a function as follows: (2)Dqfz=fqz-fzq-1zz≠0,q≠0and Dqf(z)=f′(0). In case f(z)=zk for k is a positive integer, the q-derivative of f(z) is given by (3)Dqzk=zk-zqkz1-q=kqzk-1.As q→1- and k∈N, we have (4)kq=1-qk1-q=1+q+⋯+qk⟶k.Quite a number of great mathematicians studied the concepts ofq-derivative, for example, by Gasper and Rahman [3], Aral et al. [4], Li et al. [5], and many others (see [6–15]).Making use of theq-derivative, we define the subclasses Sq∗(α) and Cq(α) of the class A for 0≤α<1 by(5)Sq∗α=f∈A:RezDqfzfz>α,z∈U,Cqα=f∈A:Re1+zqDqDqfzDqfz>α,z∈U.These classes are also studied and introduced by Seoudy and Aouf [16].Noting that(6)f∈Cqα⟺zDqf∈Sq∗α,limq→1Sq∗α=f∈A:limq→1RezDqfzfz>α,z∈U=S∗α,limq→1Cqα=f∈A:limq→1Re1+zqDqDqfzDqfz>α,z∈U=Cα,where S∗(α) and C(α) are, respectively, the classes of starlike of order α and convex of order α in U ([17, 18]).Next, we state theq-analogue of Ruscheweyh operator given by Aldweby and Darus [8] that will be used throughout.Definition 1 (see [8]).
Letf∈A. Denote by Rqλ the q-analogue of Ruscheweyh operator defined by (7)Rqλfz=z+∑k=2∞k+λ-1q!λq!k-1q!akzk,where [k]q! given by is as follows:(8)kq!=kqk-1q⋯1q,k=1,2,…;1,k=0.From the definition we observe that ifq→1, we have (9)limq→1Rqλfz=z+limq→1∑k=2∞k+λ-1q!λq!k-1q!akzk=z+∑k=2∞k+λ-1!λ!k-1!akzk=Rλfz,where Rλ is Ruscheweyh differential operator defined in [19].Using the principle of subordination andq-derivative, we define the classes of q-starlike and q-convex analytic functions as follows.Definition 2.
Forφ∈P and λ>-1, the class Sq,λ∗(φ) which consists of all analytic functions f∈A satisfies (10)zDqRqλfzRqλfz≺φz,z<1.Definition 3.
Forφ∈P and λ>-1, the class Cq,λ(φ) which consists of all analytic functions f∈A satisfies (11)1+zqDqDqRqλfzDqRqλfz≺φz,z<1,0<q<1.To prove our results, we need the following.Lemma 4 (see [18]).
Ifpz=1+c1z+c2z2+⋯∈P of positive real part is in U and μ is a complex number, then (12)c2-μc12≤2max⁡1;2μ-1.The result is sharp given by (13)pz=1+z1-z,pz=1+z21-z2.Lemma 5 (see [18]).
Ifp(z)=1+c1z+c2z2+⋯ is a function with positive real part, then (14)c2-νc12≤-4ν+2,ifν≤0;2,if0≤ν≤1;4ν-2,ifν≥1.


## 2. Main Results

Now is our theorem using similar methods studied by Seoudy and Aouf in [16].Theorem 6.
Letφz=1+B1z+B2z2+⋯∈P. If f given by (1) is in the class Sq,λ∗(φ) and μ is a complex number, then (15)a3-μa22≤B1qλq2+q2λ1+qλq2+1max1,B2B1+λq+qλ-μλq+qλ1+qqλq+qλB1.The result is sharp.Proof.
Iff∈Sq,λ∗(φ), then there is a function w(z) in U with w(0)=0 and |w(z)|<1 in U such that (16)zDqRqλfzRqλfz=φwz.Define the function p(z) by (17)pz=1+wz1-wz=1+p1z+p2z2+⋯.Since w(z) is a Schwarz function, immediately Rep(z)>0 and p(0)=1. Let (18)gz=zDqRqλfzRqλfz=1+d1z+d2z2+⋯.Then from (16), (17), and (18), obtain (19)gz=φpz-1pz+1.Since (20)pz-1pz+1=12p1z+p2-p122z2+p3+p134-p1p2z3+⋯we have (21)φpz-1pz+1=1+12B1p1z+12B1p2-p122+14B2p12z2+⋯.From the last equation and (18), we obtain (22)d1=12B1p1,d2=12B1p2-p122+14B2p12.A simple computation in (18) and knowing that [n]q-1=q[n-1]q, we obtain (23)zDqRqλfzRqλfz=1+qλ+1qa2z+qλ+1qλ+2qa3-qλ+1q2a22z2+⋯.Then, from last equation and (18), we see that (24)d1=qλ+1qa2,d2=qλ+1qλ+2qa3-qλ+1q2a22,or equivalently, we have (25)a2=B1p12qλ+1q,a3=B12qλ+1qλ+2qp2-p122+B2p124qλ+1qλ+2q+B12p128q2λ+1qλ+2q.Therefore (26)a3-μa22=B12qλ+1qλ+2qp2-νp12,where (27)ν=121-B2B1-λq+qλ-μλq+qλ1+qqλq+qλB1.By an application of Lemma 4, our result follows. Again by Lemma 4, the equality in (15) is gained for (28)pz=1+z1-zorpz=1+z21-z2.Thus Theorem 6 is complete.Similarly, we can prove for the classCq,λ(φ). We omit the proofs.Theorem 7.
Letφz=1+B1z+B2z2+⋯∈P. If f given by (1) is in the class Cq,λ(φ) and μ is a complex number, then (29)a3-μa22≤B12q3qλ+1qλ+2qmax⁡1,B2B1+2q2-μ3qλ+2qq22B1.The result is sharp.Takingλ=0 in Theorem 6, we have the corollary for the class Sq∗(φ) as follows.Corollary 8.
Letφz=1+B1z+B2z2+⋯∈P. If f given by (1) is in the class Sq∗(φ) and μ is a complex number, then (30)a3-μa22≤B1q1+qmax⁡1,B2B1+1-μ1+qqB1.The result is sharp.Takingq→1 and λ=0 in Theorem 6, we obtain the following.Corollary 9.
Letφ(z)=1+B1z+B2z2+⋯, B1∈P. If f given by (1) is in the class Sq,λ∗(φ) and μ is a complex number, then (31)a3-μa22≤B12max⁡1,B2B1+1-2μ1B1.By using Lemma4, we have the following theorem.Theorem 10.
Letφ(z)=1+B1z+B2z2+⋯ with B1>0 and B2≥0. Let (32)σ1=λq+qλB12+qB2-B1λq+qλ2qB12,σ2=λq+qλB12+qB2+B1λq+qλ2qB12.Let f given by (1) be in the class Sq,λ∗(φ). Then (33)a3-μa22≤B2qλ+1qλ+2q+B12qλ+1qλ+2qλq+qλ-λq+qλ2qμqλq+qλ,ifμ≤σ1;B1qλ+1qλ+2q,ifσ1≤μ≤σ2;-B2qλ+1qλ+2q-B12qλ+1qλ+2qλq+qλ-λq+qλ2qμqλq+qλ,ifμ≥σ2.Proof.
First, letμ≤σ1(34)a3-μa22≤B12qλ+1qλ+2q-4ν+2≤B2qλ+1qλ+2q+B12qλ+1qλ+2qλq+qλ-λq+qλ2qμqλq+qλ.Now, let σ1≤μ≤σ2; then using the above calculation, we obtain (35)a3-μa22≤B1qλ+1qλ+2q.Finally, if μ≥σ2, then (36)a3-μa22≤B1qλ+1qλ+2q4ν-2-B2qλ+1qλ+2q-B12qλ+1qλ+2qλq+qλ-λq+qλ2qμqλq+qλ.Similarly, we can prove for the classCq,λ(φ) as follows.Theorem 11.
Letφz=1+B1z+B2z2+⋯ with B1>0 and B2≥0. Let (37)ϱ1=2qB12+qB2-B13qB12,ϱ2=2qB12+qB2+B13qB12.If f given by (1) is in the class Cq,λ(φ), then (38)a3-μa22≤B12q3qλ+1qλ+2qB2B1+2q2-3qλ+2qμq2q2B1,ifμ≤ϱ1;B12q3qλ+1qλ+2q,ifϱ1≤μ≤ϱ2;B12q3qλ+1qλ+2q-B2B1-2q2-3qλ+2qμq2q2B1,ifμ≥ϱ2.Takingλ=0 in Theorem 10, we obtain next result for the class Sq∗(φ).Corollary 12.
Letφz=1+B1z+B2z2+⋯ with B1>0 and B2≥0. Let (39)σ1=B12+qB2-B12qB12,σ2=B12+qB2+B12qB12.If f given by (1) is in the class Sq∗(φ), then (40)a3-μa22≤B2q2q+B12q2q1-2qμq,ifμ≤σ1;B1q2q,ifσ1≤μ≤σ2;-B2q2q-B12q2q1-2qμq,ifμ≥σ2.



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