# Some Coefficient Inequalities of q-Starlike Functions Associated with Conic Domain Defined by q-Derivative

Authors: Shahid Mahmood; Mehwish Jabeen; Sarfraz Nawaz Malik; H. M. Srivastava; Rabbiya Manzoor; S. M. Jawwad Riaz
Journal: Journal of Function Spaces
 (2018)

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## Abstract

This article deals with q-starlike functions associated with conic domains, defined by Janowski functions. It generalizes the recent study of q-starlike functions while associating it with the conic domains. Certain renowned coefficient inequalities in connection with the previously known ones have been included in this work.


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## Body

## 1. Introduction

Quantum calculus or q-calculus is none other than a version of classical calculus because in this, we do not take limits. We consider derivatives as differences whereas antiderivatives as sums. The q-derivative of a complex valued functionf, defined in the domain D, is given as follows. (1)Dqfz=fz-fqz1-qz,z≠0,f′0,z=0,where 0<q<1. This implies the following. (2)limq→1-⁡Dqfz=limq→1-⁡fz-fqz1-qz=f′z,provided that the function f is differentiable in domain D. The function Dqf has Maclaurin’s series representation (3)Dqfz=∑n=0∞nqanzn-1,where (4)nq=1-qn1-q,n∈C,∑k=0n-1qk=1+q+q2+⋯+qn-1,n∈N.For more details about q-derivatives, we refer the reader to [1–14].We denote byA the class of functions f(z) which are analytic in the open unit disc E=z:z<1 and are of the form(5)fz=z+∑n=2∞anzn.Let S denote the class of all functions in A which are univalent in E. Also let S∗ and C be the subclasses of S consisting of all functions which map E onto a star shaped with respect to origin and convex domains, respectively. A function f is said to be subordinate to a function g, written symbolically as f≺g, if there exists a function w with w0=0,wz<1, such that fz=gwz for z∈E.The credit of systematic initiation of q-calculus goes to Jackson [13, 14] who introduced and gave the early definitions of q-derivatives and q-integrals. One of the very early contributions of usage of Q-calculus in Geometric Function Theory was made by Ismail et al. [12] who defined the generalized version of class of starlike functions. He named his newly introduced class as class of q-starlike functions since he used q-derivatives in defining it. It took a long while in further development in this direction but it proved to be a good comeback when Anastassiu and Gal [4, 5] presented their work on complex operators with their respective q-generalizations. These are known as q-Picard and q-Gauss-Weierstrass singular integral operators. Continuing this work, Srivastava [15] laid a strong foundation of application of Q-calculus in Geometric Function Theory by using basic q-hypergeometric functions. Another series of contributions was made by Aral and Gupta [6–8] who used q-beta functions to define the q-Baskakov Durrmeyer operator. They also derived a number of geometric results with their q-extensions. In the similar manner, many q-calculus operators including integral and derivative in fractional form have been used to define and analyze a number of subclasses of analytic functions. Inspired by the research carried out in q-calculus, Aldweby and Darus [1, 2] defined the q-operators by using the concept of convolution of analytic functions which are normalized. Moreover, they discussed the geometrical structure of these defined operators in the analytic functions which involve q-version of hypergeometric functions in compact disc. Several useful results related to the q-version of class of close to convex functions were proved by Sahoo and Sharma [16]. Noor et al. [17] gave the research a new direction from application point of view and derived integral inequalities for relative harmonic preinvex functions. Very recently, many researchers of Geometric Function Theory like Noor et al. [18], Ramachandran et al. [19], Altinkaya et al. [3], Bulut [9], and Mahmood and Sokół [20] have contributed to the development of results in the background of q-calculus. The work on q-polynomials and (p,q)-polynomials also contributed remarkably to the field of q-calculus; see [10, 11].In 1999, Kanas and Wiśniowska [21] introduced the concept of conic domain by defining k-uniformly convex functions and then in 2000, they defined the corresponding k-starlike functions; see [22]. The class k-ST of k-starlike functions is defined as follows.A functionfz∈A is said to be in the class k-ST if and only if (6)Rzf′zfz>kzf′zfz-1or equivalently (7)zf′zfz≺p~kz,z∈E,k≥0,where(8)p~kz=1+z1-z,k=0,1+2π2log⁡1+z1-z2,k=1,1+21-k2sinh2⁡2πarccos⁡karctan⁡hz,0<k<1,1+1k2-1sin⁡π2Rt∫0uz/t11-x21-tx2dx+1k2-1,k>1,where u(z)=(z-t)/(1-tz), t∈(0,1), z∈E, z is chosen such that k=cosh⁡πR′(t)/4R(t), R(t) is the Legendre’s complete elliptic integral of the first kind, and R′(t) is complementary integral of R(t); for more detail, see [21–25]. If p~k(z)=1+δkz+⋯, then it is shown in [26] that from (8), one can have(9)δk=8arccos⁡k2π21-k2,0≤k<1,8π2,k=1,π24k2-1t1+tR2t,k>1.This classk-ST was then generalized by Noor and Malik [27] and its generalization k-STA,B was introduced by using the concept of Janowski functions. The detail of Janowski functions may be seen from [28]. The class k-STA,B is defined as follows.A functionfz∈A is said to be in the class k-STA,B,k≥0, -1≤B<A≤1, if and only if (10)ReB-1zf′z/fz-A-1B+1zf′z/fz-A+1>kB-1zf′z/fz-A-1B+1zf′z/fz-A+1-1,or equivalently (11)zf′zfz≺A+1p~kz-A-1B+1p~kz-B-1.Motivated by the above classes, we now define the following more general class of q-starlike functions associated with conic domain defined by Janowski functions.Definition 1.
A functionpz is said to be in the class k-PqA,B, if and only if(12)pz≺A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q,k≥0,where p~kz is defined by (8), 0<q<1, and -1≤B<A≤1.Geometrically, the functionpz∈k-PA,B takes all values from the domain Ωk,qA,B, -1≤B<A≤1,k≥0 which is defined as(13)Ωk,qA,B=w:Rψ>kψ-1,ψ=B1+q-3-qwz-A1+q-3-qB1+q+3-qwz-A1+q+3-q.Definition 2.
A functionfz∈A is said to be in the class k-STqA,B, k≥0, -1≤B<A≤1, if and only if(14)RB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q>kB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q-1or equivalently(15)zDqfzfz∈k-PqA,B.It is noted that 0-STqA,B=Sq∗A,B, the well-known class of q-starlike functions, introduced by Srivastava et al. [29] and limq→1-⁡k-STqA,B=k-STA,B, the well-known class of Janowski k-starlike functions, introduced by Noor and Malik [27].


## 2. Preliminary Results

We need the following lemmas to prove our main results.Lemma 3 (see [30]).
Lethz=1+∑n=1∞cnzn be subordinate to Hz=1+∑n=1∞Cnzn. If Hz is univalent in E and HE is convex, then(16)cn≤C1,n≥1.Lemma 4 (see [31]).
Ifpz=1+p1z+p2z2+⋯ is a function with positive real part in U, then, for any real number v, (17)p2-vp12≤-4v+2,v≤0,2,0≤v≤1,4v-2,v≥1.When v<0 or v>1, the equality holds if and only if pz is (1+z)/(1-z) or one of its rotations. If 0<v<1, then, the equality holds if and only if pz=(1+z2)/(1-z2) or one of its rotations. If v=0, the equality holds if and only if (18)pz=1+η21+z1-z+1-η21-z1+z0≤η≤1,or one of its rotations. If v=1, then, the equality holds if and only if pz is reciprocal of one of the functions such that equality holds in the case of v=0. Although the above upper bound is sharp, when 0<v<1, it can be improved as follows: (19)p2-vp12+p12≤20<v≤12and (20)p2-vp12+1-vp12≤212<v≤1.


## 3. Main Results

Theorem 5.
A functionf∈A and of the form (5) is in the class k-STqA,B, if it satisfies the condition(21)∑n=2∞23-qk+1qn-1q+B1+q+3-qnq-A1+q+3-qan<1+qB-A,where -1≤B<A≤1 and k≥0.Proof.
Assuming that (21) holds, then it suffices to show that(22)kB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q-1-RB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q-1<1and we consider(23)kB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q-1-RB1+q-3-qzDqfz/fz-A1+q-3-qB1+q+3-qzDqfz/fz-A1+q+3-q-1≤k+1B1+q-3-qzDqfz-A1+q-3-qfzB1+q+3-qzDqfz-A1+q+3-qfz-1=23-qk+1fz-zDqfzB1+q+3-qzDqfz-A1+q+3-qfz=23-qk+1∑n=2∞1-nqanznB-A1+qz+∑n=2∞B1+q+3-qnq-A1+q+3-qanzn≤23-qk+1∑n=2∞1-nqan1+qB-A-∑n=2∞B1+q+3-qnq-A1+q+3-qan.The last expression is bounded above by 1 if(24)23-qk+1∑n=2∞nq-1an<1+qB-A-∑n=2∞B1+q+3-qnq-A1+q+3-qan,which reduces to (25)∑n=2∞23-qk+1qn-1q+B1+q+3-qnq-A1+q+3-qan<1+qB-Aand this completes the proof.Forq→1-, Theorem 5 reduces to the following result, proved by Noor and Malik [27].Corollary 6.
A functionf∈A and of the form (5) is in the class k-STA,B, if it satisfies the condition(26)∑n=2∞2k+1n-1+B+1n-A+1an<B-A,where -1≤B<A≤1.Theorem 7.
Let the functionf∈k-STqA,B be of the form (5); then(27)an≤∏j=0n-2A-Bq+1δk-4Bqjq4qj+1q,n∈N∖1.This result is sharp.Proof.
By definition, fork-STqA,B, we have(28)zDqfzfz=pz,where(29)pz≺A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q.If p~kz=1+δkz+⋯, then(30)A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q=1+14A-Bq+1δk+14-14Aq-14A+14Bq+14BB+11+q+2-2qδk2+⋯.Now if pz=1+∑n=1∞cnzn, then by (16) and (29), we get(31)cn≤14A-Bq+1δk.Now from (28), we have (32)zDqfz=fzpz,which implies that (33)z+∑n=2∞nqanzn=z+∑n=2∞anzn1+∑n=1∞cnzn.This implies that (34)∑n=1∞nq-1anzn=∑n=1∞∑j=1n-1an-jcjzn.Comparison of coefficients of zn gives us (35)nq-1an=∑j=1n-1an-jcj,a1=1which reduces to (36)an≤1qn-1q∑j=1n-1an-jcj.By using (31), we get(37)an≤A-Bq+1δk4qn-1q∑j=1n-1aj.Next, we need to show that(38)A-Bq+1δk4qn-1q∑j=1n-1aj≤∏j=0n-2A-Bq+1δk-4Bqjq4qj+1q.For that, we use the principle of mathematical induction. For n=2, we find from (37) that (39)a2≤A-Bq+1δk4q1q,which results also from (27). Now for n=3, we find from (37) that (40)a3≤A-Bq+1δk4q2q1+a2≤A-Bq+1δk4q2q1+A-Bq+1δk4q1q,which also follows from (27). Now let inequality (38) be true for n=m. We find from (37) that(41)am≤A-Bq+1δk4qm-1q∑j=1m-1aj.On the other hand, from (27), we have (42)am≤∏j=0m-2A-Bq+1δk-4Bqjq4qj+1q≤∏j=0m-2A-Bq+1δk+4qjq4qj+1q.By the induction hypothesis, we have(43)A-Bq+1δk4qm-1q∑j=1m-1aj≤∏j=0m-2A-Bq+1δk+4qjq4qj+1q.Multiplying both sides of (43) by (A-Bq+1δk+4qm-1q)/4qmq, we have (44)∏j=0m-1A-Bq+1δk+4qjq4qj+1q≥A-Bq+1δk+4qm-1q4qmqA-Bq+1δk4qm-1q∑j=1m-1aj=A-Bq+1δk4qmqA-Bq+1δk4qm-1q∑j=1m-1aj+∑j=1m-1aj≥A-Bq+1δk4qmqam+∑j=1m-1aj=A-Bq+1δk4qmq∑j=1maj.That is, (45)A-Bq+1δk4qmq∑j=1maj≤∏j=0m-1A-Bq+1δk+4qjq4qj+1,which shows that inequality (38) is true for n=m+1, and hence the required result.Fork=0, The above result reduces to the following result, proved by Srivastava et al. [29].Corollary 8.
Let the functionf∈Sq∗A,B be of the form (5); then(46)an≤∏j=0n-2q+1A-B-2Bqjq2qj+1q,n∈N∖1,where -1≤B<A≤1.Forq→1-, Theorem 7 reduces to the following result, proved by Noor and Malik [27].Corollary 9.
Let the functionf∈k-STA,B be of the form (5); then(47)an≤∏j=0n-2A-Bδk-2Bj2j+1,n∈N∖1,where -1≤B<A≤1.Theorem 10.
Letf∈k-STqA,B, -1≤B<A≤1,0<k<1 and be of the form (5). Then for real number μ, we have(48)a3-μa22≤A-BT24q1-k22T2+23-B+11+q+21-qT21-k2+1+qA-BT2q1-k21-μ1+q,μ≤γ1,2,γ1≤μ≤γ2,B+11+q+21-qT21-k2-2T2+23-1+qA-BT2q1-k21-μ1+q,μ≥γ2,where(49)γ1=11+q+2qT2-11-k23T21+q2A-B-qB+11+q+2q1-q1+q2A-Band(50)
γ
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.Proof.
Forp∈P and of the form pz=1+∑n=1∞cnzn, we consider (51)pz=1+wz1-wz,where wz is such that w0=0 and wz<1. It follows easily that (52)wz=pz-1pz+1=1+c1z+c2z2+c3z3+⋯-11+c1z+c2z2+c3z3+⋯+1.This implies that(53)wz=12c1z+12c2-14c12z2+12c3-12c2c1+18c13z3+⋯.If fz∈k-STqA,B,-1≤B<A≤1,0<k<1, then it follows from relation (15) that we have (54)zDqfzfz≺A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q.That is, there exists a function wz with w0=0 and wz<1 such that(55)zDqfzfz=A1+q+3-qp~kwz-A1+q-3-qB1+q+3-qp~kwz-B1+q-3-q,where p~kz=1+(2/1-k2)sinh22/πarccos⁡karctan⁡hz,0<k<1. Now if p~kz=1+R1kz+R2kz2+R3kz3+⋯, then from (53), one may have (56)p~kwz=1+R1kwz+R2kw2z+R3kw3z+⋯=1+R1k12c1z+12c2-14c12z2+12c3-12c2c1+18c13z3+⋯+R2k12c1z+12c2-14c12z2+12c3-12c2c1+18c13z3+⋯2+R3k12c1z+12c2-14c12z2+12c3-12c2c1+18c13z3+⋯3+⋯where R1k,R2k, and R3k are given by (57)R1k=2T21-k2,R2k=2T231-k22+T2,R3k=2T291-k2235+4T2+25T4and T=Tk=(2/π)arccos⁡k,0≤k≤1; see [26]. Using these, the above series reduces to(58)p~kwz=1+T21-k2c1z+T21-k2T2-116c12+c2z2+T21-k21925-12T2+110T4c13-131-T2c2c1+c3z3+⋯.Using this, (55) becomes(59)zDqfzfz=1+A-B1+qT2c141-k2z+A-B1+qT281-k2T2-13-T2B+Bq+3-q21-k2c12+A-B1+qT241-k2c2z2+A-B1+qT241-k2c3+-T2-13-T2B+Bq+3-q21-k2c1c2+245-T218+T490-T2T2-1B+Bq+3-q241-k2+B+Bq+3-qT2481-k222T2k2-3T2q+7T2+2-2k2+3T2B+3T2Bqc13z3+⋯.If Dqf(z)=1+∑2∞nqanzn and fz=z+∑2∞anzn, then(60)zDqfzfz=z+∑2∞nqanznz+∑2∞anzn=1+qa2z+q+q2a3-qa22z2+qa23-a2a32q+q2+a4q+q2+q3z3+⋯.From (59) and (60), comparison of coefficients of z and z2 gives(61)a2=1+qA-BT2c14q1-k2,and(62)a3=1+qT4A-B216q21-k22+A-BT2T2-124q1-k2-T4A-B16q1-k22B+11+q+21-qc12+T2A-B4q1-k2c2.Now for a real number μ, we consider (63)a3-μa22=1+qT4A-B216q21-k22+A-BT2T2-124q1-k2-T4A-BB+11+q+21-q16q1-k22c12+T2A-B4q1-k2c2-μ1+q2T4A-B216q21-k22c12=T2A-B4q1-k2c2--1+qT2A-B4q1-k2-T2-16+T2B+11+q+21-q41-k2+μ1+q2T2A-B4q1-k2c12=T2A-B4q1-k2c2-νc12,where (64)ν=T241-k2B+11+q+21-q-T2-16-1+qT2A-B4q1-k21-μ1+q.Applying Lemma 4, we get the required result. Inequality (48) is sharp and equality holds for μ<γ1 or μ>γ2 when fz is f1z or one of its rotations, where f1z is defined such that (65)zDqf1zf1z=A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q.If γ1<μ<γ2, then, the equality holds for the function f2z or one of its rotations, where f2z is defined such that (66)zDqf2zf2z=A1+q+3-qp~kz2-A1+q-3-qB1+q+3-qp~kz2-B1+q-3-q.If μ=γ1, the equality holds for the function f3z or one of its rotations, where f3z is defined such that (67)zDqf3zf3z=1+η2A1+q+3-qp~kz-A1+q-3-qB1+q+3-qp~kz-B1+q-3-q+1-η2A1+q+3-qp~k-z-A1+q-3-qB1+q+3-qp~k-z-B1+q-3-q,where 0≤η≤1. If μ=γ2, then, the equality holds for fz, which is such that zDqf(z)/f(z) is reciprocal of one of the function such that equality holds in the case of μ=γ1, where γ1 and γ2 are defined by (49) and (50), respectively.Fork=0, the above result reduces to the following result, proved by Srivastava et al. [29].Corollary 11.
Letf∈Sq∗A,B, -1≤B<A≤1 and be of the form (5). Then for real number μ, we have(68)a3-μa22≤A-B4q22q-B+11+qq-21-qq+1+qA-B1-μ1+q,μ≤γ3,2q,γ3≤μ≤γ4,B+11+qq+21-qq-2q-1+qA-B1-μ1+q,μ≥γ4,where (69)γ3=11+q-B+11+qq+21-qq1+q2A-Band (70)γ4=11+q-B+11+qq+21-qq1+q2A-B+4q1+q2A-B.Inequality (68) is sharp.Forq→1-, Theorem 10 reduces to the following result.Corollary 12.
Letf∈k-STA,B, -1≤B<A≤1,0<k<1 and be of the form (5). Then for real number μ, we have(71)a3-μa22≤A-BT241-k22T2+23-2B+1T21-k2+2T2A-B1-k21-2μ,μ≤γ5,2,γ5≤μ≤γ6,2B+1T21-k2-2T2+23-2T2A-B1-k21-2μ,μ≥γ6,where (72)γ5=12+T2-11-k26T2A-B-B+12A-Band (73)γ6=12+T2-11-k26T2A-B-B+12A-B+1-k2T2A-B.Inequality (71) result is sharp.Theorem 13.
Letf∈1-STqA,B, -1≤B≤A≤1 and be of the form (5). Then for real number μ, we have(74)a3-μa22≤A-Bqπ243-4π221-q+B+1q+1+4A-B1+qqπ21-q+1μ,μ≤α1,2,α1≤μ≤α2,-43+4π221-q+B+1q+1-4A-B1+qqπ21-1+qμ,μ≥α2,where (75)α1=11+q-qπ26A-B1+q2-2q1-q+qB+1q+1A-B1+q2and (76)α2=11+q+5qπ261+q2A-B-2q1-q+qB+1q+1A-B1+q2.Inequality (74) is sharp.Proof.
The use of functionp~z=1+(2/π2)log⁡1+z/1-z instead of (77)p~kz=1+21-k2sinh2⁡2πarccos⁡karctan⁡hz,0<k<1,in the proof of Theorem 10 and following the similar method as followed in Theorem 10 will lead us to the required result.Forq→1-, the theorem reduces to the following form.Corollary 14.
Letf∈STA,B, -1≤B<A≤1 and be of the form (5). Then, for a real number μ,(78)a3-μa22≤A-Bπ243-8π2B+1+8π2A-B1-2μ,μ≤12-π224A-B-B+12A-B,------------2,12-π224A-B-B+12A-B≤μ≤12+5π224A-B-B+12A-B,-------------43+8π2B+1-8π2A-B1-2μ,μ≥12+5π224A-B-B+12A-B.This result is sharp.Now we consider the inverse functionF, defined as Fw=Ffz=z, z∈U, and we find the following coefficient bound for inverse functions.Theorem 15.
Letw=fz∈k-STqA,B, -1≤B<A≤1,0<k<1, and Fw=f-1w=w+∑n=2∞dnwn. Then, (79)dn≤A-B1+qT22q1-k2∑k=0n-2qkn=2,3,4.Proof.
SinceFw=Ffz=z, it is easy to see that (80)d2=-a2,d3=2a22-a3,d4=-a4+5a2a3-5a23.By using (61) and (62), one can have (81)d2=-1+qA-BT2c14q1-k2and(82)d3=A-B21+q2T48q21-k22-A-B21+qT416q21-k22-A-BT2T2-124q1-k2+A-BT4B+Bq+3-q16q1-k22c12+-A-BT24q1-k2c2.From (59) and (60), comparison of z3 gives(83)d4=-A-B1+qT24q+q2+q31-k2c3+T23-A-B5q2+4q+3T24q1-k2-13-T2B+Bq+3-q21-k2c2c1+A-B21+q5q3+5q2+5q+1T416q21-k22-A-B5q2+4q+3T2T2-124q1-k22+A-B5q2+4q+3T4B+Bq+3-q16q1-k22-T2T2-1B+Bq+3-q241-k2+245-T218+T490+T2B+Bq+3-q481-k222T2k2-3T2q+7T2+2-2k2+3T2B+3T2Bqc13.Now, from (81) and (82), one can have (84)d2≤A-B1+qT22q1-k2and (85)d3≤A-BT24q1-k2λ1c2-c12+λ2c2,where (86)λ1=16+A-B1+q2T22q1-k2-1+qT2A-B4q1-k2-T26+T2B+Bq+3-q41-k2and (87)λ2=56-A-B1+q2T22q1-k2+1+qT2A-B4q1-k2+T26-T2B+Bq+3-q41-k2.Application of the bounds c2-c12≤2 and c2≤2 (see Lemma 4 for v=1 and v=0) gives d3≤A-BT2/2q1-k2. Lastly, (83) reduces to(88)d4≤A-B1+qT24q+q2+q31-k2λ3c3-2c2c1+c13+λ4c3-c2c1+λ5c3,where (89)
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2and(90)λ5=3245-5T26-13+A-B5q2+4q+34q1-k22+B+Bq+3-q21-k2+T4A-B21+q5q3+5q2+5q+116q21-k22-A-B5q2+4q+324q1-k22+A-B5q2+4q+3B+Bq+3-q16q1-k22-B+Bq+3-q241-k2+190+T2B+Bq+3-q241-k2-1+2T2k2-3T2q+7T2+2-2k2+3T2B+3T2Bq21-k2.Applying the bounds c3-2c2c1+c13≤2, see [32], c3-c2c1≤2 and c3≤2, see [31], to the right hand side of (88) and using the fact that λi≥0,i=1,2,3, we have d4≤A-B1+qT2/21-k2q+q2+q3 and this completes the proof.Theorem 16.
Letw=fz∈k-STqA,B, -1≤B<A≤1,0<k<1, and Fw=f-1w=w+∑n=2∞dnwn. Then for real number μ, we have (91)d3-μd22≤A-BT24q1-k243+2T23-T2B+Bq+3-q1-k2-T2A-Bq+1q1-k21+2q-μ1+q,μ≤α3,2,α3≤μ≤α4,-43-2T23+T2B+Bq+3-q1-k2+T2A-Bq+1q1-k21+2q-μ1+q,μ≥α4,where (92)α3=1+2q1+q+2q1-k231+q2A-BT2-2q1-k231+q2A-B+qB+Bq+3-q1+q2A-Band (93)α4=1+2q1+q-10q1-k231+q2A-BT2-2q1-k231+q2A-B+qB+Bq+3-q1+q2A-B.Proof.
The proof follows directly from (81), (82), and Lemma 4.Following the similar method as proceeded in Theorems15 and 16, we have the following results for k=1.Theorem 17.
Letw=fz∈1-STqA,B, -1≤B<A≤1, and Fw=f-1w=w+∑n=2∞Mnwn. Then, (94)Mn≤2A-Bq+1qπ2∑k=0n-2qkn=2,3,4.Theorem 18.
Letw=fz∈1-STqA,B, -1≤B<A≤1, and Fw=f-1w=w+∑n=2∞Mnwn. Then for real number μ, we have (95)M3-μM22≤A-Bqπ243-4q+1B+Bq+3-qπ2-4A-Bqπ22q2+3q+12-1+q2μ,μ≤α5,2,α5≤μ≤α6,-43+4q+1B+Bq+3-qπ2+4A-Bqπ22q2+3q+12-1+q2μ,μ≥α6.where (96)α5=2q2+3q+11+q2+qπ26A-Bq+12+qB+Bq+3-qA-Bq+1and (97)α6=2q2+3q+11+q2-5qπ26A-Bq+12+qB+Bq+3-qA-Bq+1.


## 4. Conclusion

We have generalized the concept of q-starlike functions by associating them with conic domain. The concept of q-derivatives is used to define class of certain analytic functions. The certain renowned coefficient inequalities are found for these functions and their inverse functions.



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*Source: 8492072-2018-10-08.xml*