# Certain Bound forq-Starlike and q-Convex Functions with respect to Symmetric Points

Authors: C. Ramachandran; D. Kavitha; T. Soupramanien
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

The aim of this paper is to establish the coefficient estimates for the subclasses ofq-starlike and q-convex functions with respect to symmetric points involving q-difference operator. Also certain applications based on these results for subclasses of univalent functions defined by convolution are given.


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## Body

## 1. Introduction

Theq-difference calculus or quantum calculus was initiated at the beginning of 19th century that was initially developed by Jackson [1, 2]. Basic definitions and properties of q-deference calculus can be found in the book mentioned in [3]. The fractional q-difference calculus had its origin in the works by Al-Salam [4] and Agarwal [5]. Recently, the area of q-calculus has attracted the serious attention of researchers. The great interest is due to its application in various branches of mathematics and physics. Mohammed and Darus [6] studied approximation and geometric properties of these q-operators for some subclasses of analytic functions in compact disk.LetA denote the class of all functions f(z) of the form(1)fz=z+∑n=2∞anznwhich are analytic in the open unit disk U={z∈C:z<1}.LetS be the subclass of A consisting of all univalent functions in U.Iff(z) and g(z) are analytic in U, then we say that the function f(z) issubordinate to g(z), if there exists a Schwarz function w(z), analytic in U with (2)w0=0,wz<1z∈U,such that(3)fz=gwzz∈U.We denote this subordination by (4)f≺gorfz≺gzz∈U.In particular, if the function g(z) is univalent in U, the above subordination is equivalent to f0=g0 and f(U)⊆g(U) (see [7, 8]).Aq-analog of the class of starlike functions was first introduced in [9] by means of the q-difference operator Dqf(z) acting on functions f∈A given by (1) and 0<q<1, and the q-derivative of a function f(z) is defined by (see [1, 2])(5)Dqfz=fz-fqz1-qzz≠0,Dqf(0)=f′(0) and Dq2f(z)=Dq(Dqf(z)). From (5), we deduce that(6)Dqfz=1+∑k=2∞kqakzk-1,where(7)kq=1-qk1-q.As q→1-, [k]q→k. For a function h(z)=zk, we observe that (8)Dqhz=Dqzk=1-qk1-qzk-1=kqzk-1,limq→1-⁡Dqhz=limq→1-⁡kqzk-1=kzk-1=h′z,where h′ is the ordinary derivative.As a right inverse, Jackson [2] introduced the q-integral (9)∫0zftdqt=z1-q∑k=0∞qkfzqk,provided that the series converges. For a function h(z)=zk, we have (10)∫0zhtdqt=∫0ztkdqt=zk+1k+1qk≠-1limq→1-⁡∫0zhtdqt=limq→1-⁡zk+1k+1q=zk+1k+1=∫0zhtdt,where ∫0z‍h(t)dt is the ordinary integral. Note that the q-difference operator plays an important role in the theory of hypergeometric series and quantum physics (see, e.g., [10–14]).Ma and Minda [15] unified various subclasses of starlike and convex functions for which either quantity zf′(z)/f(z) or quantity 1+zf′′(z)/f′(z) is subordinate to a more general superordinate function. For this purpose, they considered an analytic function ϕ with positive real part in the unit disc U, with ϕ(0)=1, ϕ′(0)>0, and ϕ maps U onto a region starlike, with respect to the real axis.The class of Ma-Minda starlike functions consists of functionsf(z)∈A satisfying the subordination zf′(z)/f(z)≺ϕ(z). Similarly, the class of Ma-Minda convex functions consists of functions f∈A satisfying the subordination 1+zf′′(z)/f′(z)≺ϕ(z).Sakaguchi [16] introduced and studied class Ss∗ of starlike functions with respect to symmetric points. Class Sq,s∗(ϕ) defined below is the generalization of class Ss∗.Definition 1 (see [17]).
Letϕ(z)=1+B1z+B2z2+B3z3+⋯ be a univalent starlike function with respect to 1 which maps the unit disk U onto a region in the right half plane which is symmetric with respect to the real axis and let B1>0. The function f∈A is in classes Ss∗(ϕ) and Cs(ϕ): (11)2zf′zfz-f-z≺ϕz,2zf′z′f′z+f′-z≺ϕz.Definition 2.
A functionf∈A is said to belong to classes Sq,s∗(ϕ) and Cq,s(ϕ) if (12)2zDqfzfz-f-z≺ϕz,2DqzDqfzDqfz+Dqf-z≺ϕz,respectively.Asq→1-, we denote subclasses Sq,s∗(ϕ) and Cq,s(ϕ) by Ss∗(ϕ) and Cs(ϕ) which are the class of starlike and convex functions with respect to symmetric points, respectively (see [18]).The main objective of the present paper is to derive the Fekete-Szegö inequality for functions in classesSq,s∗(ϕ) and Cq,s(ϕ). Also, by using convolution product, we give the applications of our results to the defined functions and in particular we consider classes Sq,sλ(ϕ) and Cq,sλ(ϕ) defined by fractional derivatives.In order to prove the main result, we need the following lemma.Lemma 3 (see [15]).
Ifp(z)=1+c1z+c2z2+⋯ is an analytic function with positive real part in U, then(13)c2-νc12≤-4ν+2ifν≤02if0≤ν≤14ν-2ifν≥1.
Whenν<0 or ν>1, the equality holds if and only if p(z)=(1+z)/(1-z) or one of its rotations. When 0<ν<1, then the equality holds if and only if p(z)=(1+z2)/(1-z2) or one of its rotations. If ν=0, the equality holds if and only if p(z)=1/2+λ/2(1+z/1-z)+1/2-λ/2((1-z)/(1+z)), (0≤λ≤1) or one of its rotations. While ν=1, equality holds if and only if p(z) is the reciprocal of one of the functions for which the equality holds in the case of ν=0. Also the above upper bound can be improved as follows: when 0<ν<1(14)c2-νc12+νc12≤20<ν≤12,c2-νc12+1-νc12≤212<ν≤1.We also need the following results in our investigation.Lemma 4 (see [19]).
Ifp(z)=1+c1z+c2z2+⋯ is an analytic function with positive real part in U, then (15)c2-νc12≤2max⁡1,2ν-1.The result is sharp for the function p(z)=(1+z2)/(1-z2) and p(z)=(1+z)/(1-z).


## 2. Main Results

Theorem 5.
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Iff(z) given by (1) belongs to Sq,s∗(ϕ), then(16)a3-μa22≤12B2-μB122q23q-1ifμ≤σ1,B12ifσ1≤μ≤σ2,-12B2-μB122q23q-1ifμ≥σ2,where(17)σ1≔2q23q-1B2-B1B12,σ2≔2q23q-1B1+B2B12.The result is sharp.Proof.
Forf∈Sq,s∗(ϕ), let(18)pz=2zDqfzfz-f-z=1+b1z+b2z2+⋯.From (18), we obtain (19)2qa2=b1,3q-1a3=b2.Since ϕ(z) is univalent and p≺ϕ, the function (20)p1z=1+ϕ-1pz1-ϕ-1pz=1+c1z+c2z2+⋯is analytic and has positive real part in U. Thus we have(21)pz=ϕp1z-1p1z+1and from (18) and (21), (22)a2=122qB1c1,a3=13q-1B12c2-c122+B2c124a3-μa22=12B13q-1c2-νc12,where (23)ν=12-B22B1+μB122q23q-1.Our result now follows by an application of Lemma 3. To show that these bounds are sharp, we define the functions Kϕn(n=2,3,…) by (24)2zKϕn′zKϕnz-Kϕn-z=ϕzn-1,Kϕn0=0=Kϕn′0-1and the functions Fλ and Gλ(0≤λ≤1) by (25)2zFλ′zFλz-Fλ-z=ϕzz+λ1+λz,Fλ0=0=Fλ′0-1,2zGλ′zGλz-Gλ-z=ϕ-zz+λ1+λz,Gλ0=0=Gλ′0-1.
Clearly the functionsKϕn,Fλ,Gλ∈Sq,s∗(ϕ). Also we write Kϕ≔Kϕ2. If μ<σ1 or μ>σ2, the equality holds if and only if f is Kϕ or one of its rotations. When σ1<μ<σ2, the equality holds if and only if f is Kϕ3 or one of its rotations. If μ=σ1 then the equality holds if and only if f is Fλ or one of its rotations. If μ=σ2 then the equality holds if and only if f is Gλ or one of its rotations.
Ifσ1≤μ≤σ2, then, in view of Lemma 3, Theorem 5 can be improved.Theorem 6.
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Letf(z) given by (1) belongs to Sq,s∗(ϕ). Let σ3 be given by σ3≔[2]q2B2/[3]q-1B12.
Ifσ1≤μ≤σ3, then (26)a3-μa22+1B123q-12q2B1-B2+μB123q-1a22≤B12.If σ3≤μ≤σ2, then (27)a3-μa22+1B123q-12q2B1+B2-μB123q-1a22≤B12.Corollary 7 (see [18]).
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Iff(z) given by (1) belongs to Sq,s∗(ϕ) and q→1- (i.e., Sq,s∗(ϕ) reduced to Ss∗(ϕ)) then(28)a3-μa22≤12B2-μB122ifμ≤σ1,B12ifσ1≤μ≤σ2,-12B2-μB122ifμ≥σ2,where (29)σ1≔2B2-B1B12,σ2≔2B2+B1B12.The result is sharp.Theorem 8.
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Iff(z) given by (1) belongs to Cq,s(ϕ), then(30)a3-μa22≤16B2-3q22q2μB12ifμ≤σ1,B16ifσ1≤μ≤σ2,-16B2-3q22q2μB12ifμ≥σ2,where(31)σ1≔22q23qB2-B1B12,σ2≔22q23qB1+B2B12.The result is sharp.Proof.
Let(32)pz=2DqzDqfzDqfz+Dqf-z=1+b1z+b2z2+⋯.
Sinceϕ(z) is univalent and p≺ϕ, the function (33)p1z=1+ϕ-1pz1-ϕ-1pz=1+c1z+c2z2+⋯is analytic and has positive real part in U. Thus we have(34)pz=ϕp1z-1p1z+1and from (34), (35)a2=12B1c12q,a3=B122q3qc2-c122+B2c122B1,a3-μa22=12B12q3qc2-νc12,where (36)ν=12-B22B1+μB13q22q.Our result now follows by an application of Lemma 3. To show that these bounds are sharp, we define the functions Kϕn(n=2,3,…) by (37)2Kϕn′zKϕn′zKϕn′z+Kϕn′-z=ϕzn-1,Kϕn0=0=Kϕn′0-1and the functions Fλ and Gλ(0≤λ≤1) by (38)2Fλ′zFλ′zFλ′z+Fλ′-z=ϕzz+λ1+λz,Fλ0=0=Fλ′0-1,2Gλ′zGλ′zGλ′z+Gλ′-z=ϕ-zz+λ1+λz,Gλ0=0=Gλ′0-1.Clearly the functions Kϕn,Fλ,Gλ∈Cq,s(ϕ). Also we write Kϕ≔Kϕ2. If μ<σ1 or μ>σ2, the equality holds if and only if f is Kϕ or one of its rotations. When σ1<μ<σ2, the equality holds if and only if f is Kϕ3 or one of its rotations. If μ=σ1 then the equality holds if and only if f is Fλ or one of its rotations. If μ=σ2 then the equality holds if and only if f is Gλ or one of its rotations.
Ifσ1≤μ≤σ2, then, in view of Lemma 3, Theorem 8 can be improved.Corollary 9.
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Letf(z) given by (1) belong to Cq,s(ϕ). Let σ3 be given by σ3≔(2[2]q2/[3]q)(B2/B12).
Ifσ1≤μ≤σ3, then(39)a3-μa22+22q23q1B12B1-B2+3q22q2μB12≤B16.If σ3≤μ≤σ2, then (40)a3-μa22+22q23q1B12B1+B2-3q22q2μB12≤B16.Corollary 10 (see [18]).
Let0≤μ≤1. Also let ϕ(z)=1+B1z+B2z2+B3z3+⋯, where the coefficients Bn are real with B1>0 and B2≥0.
Iff(z) given by (1) belongs to Cq,s(ϕ) and q→1- (i.e., Cq,s(ϕ) is reduced to Cs(ϕ)) then(41)a3-μa22≤16B2-38μB12ifμ≤σ1,B16ifσ1≤μ≤σ2,-16B2-38μB12ifμ≥σ2,where(42)σ1≔2B2-B1B12,σ2≔2B2+B1B12.The result is sharp.


## 3. Applications to Functions Defined by Fractional Derivative

In order to introduce classesSq,sλ(ϕ) and Cq,sλ(ϕ) we need the following.Definition 11 (see [20]).
Letf(z) be analytic in a simply connected region of the Z-plane containing the origin. The fractional derivative of f of order λ is defined by (43)Dzλfz≔1Γ1-λddz∫0zfζz-ζλdζ0≤λ<1,where the multiplicity of (z-ζ)-λ is removed by requiring that log(z-ζ) is real for z-ζ>0. Using the above Definition 11 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [20] introduced the operator Ωλ:A→A defined by (44)Ωλfz=Γ2-λzλDzλfz,λ≠2,3,4,….Classes Sq,sλ(ϕ) and Cq,sλ(ϕ) consist of functions f∈A for which Ωλf∈Sq,s∗(ϕ) and Ωλf∈Cq,s(ϕ), respectively.For a fixedg∈A, let Sq,sg(ϕ) be the class of functions f∈A for which (f∗g)∈Sq,s∗(ϕ) and let Cq,sg(ϕ) be the class of functions f∈A for which (f∗g)∈Cs(ϕ).ClassesSq,sλ(ϕ) and Cq,sλ(ϕ) are the special case of classes Sq,sg(ϕ) and Cq,sg(ϕ), respectively, when(45)gz=z+∑n=0∞Γn+1Γ2-λΓn+1-λzn.Let (46)gz=z+∑n=2∞gnzngn>0,fz=z+∑n=2∞anzn.Sincef∈Sq,sg(ϕ)Cq,sg(ϕ) if and only if f∗g∈Sq,s∗(ϕ)Cq,s(ϕ), we obtain the coefficient estimates for functions in classes Sq,sg(ϕ) and Cq,sg(ϕ), from the corresponding estimates for functions in classes Sq,s∗(ϕ) and Cq,s(ϕ).Applying Theorem5 for the function (f∗g)(z)=z+g2a2z2+g3a3z3+⋯, we get the following theorem for an obvious change of μ.Theorem 12.
Let the functionϕ(z) be given by ϕ(z)=1+B1z+B2z2+⋯ and let g(z)=z+∑n=2∞‍gnzn, (gn>0). If f(z) given by (1) belongs to Sq,sg(ϕ), then(47)a3-μa22≤12g3B2-3q-12q2μg3g22B12ifμ≤σ1,B12g3ifσ1≤μ≤σ2,-12g3B2-3q-12q2μg3g22B12ifμ≥σ2,where(48)σ1≔2q2g223q-1g3B2-B1B12,σ2≔2q2g223q-1g3B1+B2B12.The result is sharp.Since(49)Ωλfz=z+∑n=0∞Γn+1Γ2-λΓn+1-λanzn,we have(50)g2≔Γ3Γ2-λΓ3-λ=22-λ,g3≔Γ4Γ2-λΓ4-λ=62-λ3-λ.Theorem 13.
Let the functiong(z)=z+∑n=2∞‍gnzn, (gn>0). If f(z) given by (1) belongs to Cq,sg(ϕ), then(51)a3-μa22≤16g3B2-3q22q2μg3g22B12ifμ≤σ1,B16g3ifσ1≤μ≤σ2,-16g3B2-3q22q2μg3g22B12ifμ≥σ2,where (52)σ1≔22q2g223qg3B2-B1B12,σ2≔22q2g223qg3B1+B2B12.The result is sharp.Forg2 and g3 given by (50), Theorems 12 and 13 reduce to the following.Corollary 14.
Letλ<2. If f(z) given by (1) belongs to Sq,sλ(ϕ), then(53)a3-μa22≤2-λ3-λ12B2-ξμB12ifμ≤σ1,2-λ3-λ12B1ifσ1≤μ≤σ2,-2-λ3-λ12B2-ξμB12ifμ≥σ2,where (54)ξ=32-λ3q-123-λ2q2σ1≔23-λ2q232-λ3q-1B2-B1B12,σ2≔23-λ2q232-λ3q-1B1+B2B12.Corollary 15.
Letλ<2. If f(z) given by (1) belongs to Cq,sλ(ϕ), then(55)a3-μa22≤2-λ3-λ36B2-33q2-λ42q23-λμB12ifμ≤σ1,2-λ3-λ36B1ifσ1≤μ≤σ2,-2-λ3-λ36B2-33q2-λ42q23-λμB12ifμ≥σ2,where (56)σ1≔22q3-λ33q2-λB2-B1B12,σ2≔22q3-λ33q2-λB1+B2B12.



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