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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(1): 75-98

Published online March 31, 2021

### Univalent Functions Associated with the Symmetric Points and Cardioid-shaped Domain Involving (p,q)-calculus

Om Ahuja, Nisha Bohra, Asena Çetinkaya, Sushil Kumar*

Department of Mathematical Sciences, Kent State University, Ohio, 44021, U.S.A
e-mail : oahuja@kent.edu

Department of Mathematics, Sri Venkateswara College, University of Delhi, Delhi-110 021, India
email : nishib89@gmail.com

Department of Mathematics and Computer Science, İstanbul Kültür University, İstanbul, Turkey
e-mail : asnfigen@hotmail.com

Bharati Vidyapeeth's college of Engineering, Delhi-110063, India
e-mail : sushilkumar16n@gmail.com

Received: March 19, 2020; Revised: July 28, 2020; Accepted: July 28, 2020

In this paper, we introduce new classes of post-quantum or (p,q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p,q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p,q)-logarithmic coefficients. In addition, we get q-Bieberbach-de-Branges type inequalities for the special case of our classes when p=1. Moreover, we also discuss some special cases of the obtained results.

Keywords: (p, q)-Fekete-Szegö, inequalities, (p,q)-starlike functions, (p,q)-convex functions, (p,q)-logarithmic coefficient bounds, cardioid-shaped domain, q-Bie

We first recall the basic definitions in univalent function theory. Let $A$ denote the class of analytic functions f in the open unit disk $D={z∈ℂ:|z|<1}$ normalized by the conditions f(0)=0 and f'(0)=1. If $f∈A$, then

$f(z)=z+∑ n=2∞anzn (z∈D).$

Let $S$ be the subclass of $A$ containing all the univalent functions in $D$. A function $f∈A$ is starlike if and only if RE $zf′(z)/f(z)>0$ and convex if and only if RE$1+zf″(z)/f′(z)>0$ for all $z∈D$. We denote these two classes, respectively, by $S*$ and $K$. Closely related with these two classes is the class $P$ of all functions $ψ$ which are analytic and have positive real part in $D$ with $ψ(0)=1$. An analytic function f is subordinate to an analytic function g, denoted by $f≺g$, if there is an analytic function w defined on $D$ with w(0)=0 and $|w(z)|<1$ satisfying $f=g°w$. By making use of subordination, Kumar and Ravichandran [17] introduced and studied the geometric properties of the class $SR*=S*(φ0)$, where $φ0$ is a rational function given by

$φ0(z):=1+zkk+zk−z=1+1kz+2k2z2+2k3z3+⋯, k=2+1.$

Note that the image of the open unit disk under the rational function $φ0(z)$ is Cardioid shaped bounded region as shown in Figure 1. Chakrabarti and Jagannathan [8] in 1991 introduced the concept of (p,q)-calculus in order to generalize or unify several forms of q-oscillator algebras well-known in the physics literature. We first recall some basics of (p,q)-calculus. Let $0. The (p,q)-bracket or twin basic number $[n]p,q$ is defined by

Figure 1. Image of unit disk $D$ under the function $φ0$
$[n]p,q=pn−qnp−q,if q≠p,npn−1,if q=p.$

For $0, q-bracket [n]q for $n=0,1,2,…$ is given by $[n]q=[n]1,q$. The (p,q)-derivative of a function f is defined by

$Dp,qf(z)=f(pz)−f(qz)(p−q)z,if p≠q,z≠0,1,if p≠q,z=0,f′(z),if p=q.$

In particular, $Dp, qzn=[n]p, qzn−1$. For a function $f∈A$, the (p,q)-derivative operator is given by $Dp,qf(z)=1+∑ n=2∞[n]p,qanzn−1$. For definitions and properties of (p,q)-calculus, one may refer to [8]. The (1,q) derivative operator D1,q, denoted by Dq is known as the q-derivative operator. Jackson in 1909 and 1910 [14, 15], initiated the study of q-operator Dq defined by

$Dqf(z)=f(z)−f(qz)(1−q)z,if 0

Quantum Calculus or q-calculus is a theory of calculus where smoothness is not required. It may be considered as an extension of classical analysis discovered by Newton and Leibniz. In recent years, q-calculus has attracted attention of many researchers due to its wide range of applications in various fields, as for example, in the areas of ordinary fractional calculus, orthogonal polynomials, basic hypergeometric functions, combinatorics, the calculus of variations and many others. For further details about the theory of q-calculus, one may refer to [2, 3, 5, 7, 13, 21].

If $f,g∈A$ with

$f(z)=∑ n=1∞anzn and g(z)=∑ n=1∞bnzn,$

then the convolution or Hadamard product of f and g, denoted by $f∗g$, is defined by

$f(z)∗g(z)=(f∗g)(z)=z+∑ n=2∞anbnzn.$

If $g(z)=z/(1−z)$, then obviously $f∗g=f$ for all $f∈A$. Again, if $g(z)=z/(1−z)2,$ then it is straightforward to notice that $f∗g=zf′$ for all $f∈A$.

We now introduce two new classes of (p,q)-starlike and (p,q)-convex functions with respect to symmetric points associated with the rational function $φ0$.

### Definition 1.1.

A function $f∈S$ is said to be in class $STSp,qφ0(z)$ if it satisfies the following subordination condition:

$2zDp,qf(z)f(z)−f(−z)≺φ0(z), (z∈D).$

where $φ0$ is a rational function defined by (1.2).

The functions in the class $STSp,qφ0(z)$ may be called as (p,q)-starlike functions with respect to symmetric points associated with the rational function $φ0$. Using the concept of the subordination, we say that there exists a function w analytic in $D with w(0)=0,|w(z)|<1$ such that

$2zDp,qf(z)f(z)−f(−z)=1+w(z)(k+w(z))k(k−w(z)),$

which is equivalent to

$2zDp,qf(z)f(z)−f(−z)≠k2+e2iθk2−keiθ, (z∈D;θ∈[0,2π])$

or

$2(k2−keiθ)zDp,qf(z)−(k2+e2iθ)(f(z)−f(−z))≠0$

In view of convolution, we obtain

$f(z)−f(−z)=f(z)∗z1−z−f(z)∗−z1−(−z)=2f(z)∗z(1−z)(1+z)$

and

$zDp,qf(z)=f(z)∗z(1−pz)(1−qz).$

On substituting (1.5) and (1.6) into (1.4), we conclude that the function f is in the class $STSp,qφ0(z)$ if and only if

$1z[f(z)∗(2(k2−keiθ)z(1−pz)(1−qz)−2(k2+e2iθ)z(1−z)(1+z))]≠0.$

We remark that for technique used in proving (1.7), one may refer to [1, 30].

### Remark 1.2.

If we replace the rational function $φ0$ in Definition 1.1 by the function (1+z)/(1-z), then the subordination condition (1.3) reduces to the inequality

$Re2zDp,qf(z)f(z)−f(−z)>0, (z∈D);$

and it gives rise to a new class of (p,q)-starlike functions with respect to symmetric points, denoted by $Ss*(p,q)$. For p=1, the class $Ss*(q)≡Ss*(1,q)$ is a special case of a more general class $Ss*(ϕ;p,q)$ where ϕ is an analytic function with positive real part in $D$ with $ϕ(0)=1,ϕ′(0)>0$. The class $Ss*(ϕ;1,q)$ was studied in [24]. Also, note that for p=1 and $ϕ(z)=(1+z)/(1−z)$, the class

$Ss*≡limq→1−Ss*1+z1−z;1,q=f∈S:Re2zf′(z)f(z)−f(−z)>0, z∈D$

was introduced and studied by Sakaguchi [26].

### Definition 1.3.

A function $f∈S$ is said to be in class $CSp,qφ0(z)$ if the following subordination condition holds:

$2Dp,q(zDp,qf(z))Dp,q(f(z)−f(−z))≺φ0(z), (z∈D).$

where $φ0$ is a rational function defined by (1.2).

The functions in the class $CSp,qφ0(z)$ may be called as (p,q)-convex functions with respect to symmetric points associated with the rational function $φ0$. Applying the method used in proving (1.7), we find that the function f is in the class $CSp,qφ0(z)$ if and only if

$1z[f(z)∗(2(k2−keiθ)(z+pqz2)(1−p2z)(1−q2z)(1−pqz)−2(k2+e2iθ)(z+pqz3)(1−p2z2)(1+q2z2))]≠0$

### Remark 1.4.

If we replace the rational function $φ0$ in Definition 1.3 by the function (1+z)/(1-z), then the subordination condition (1.8) reduces to the inequality

$Re2Dp,q(zDp,qf(z))Dp,q(f(z)−f(−z))>0, (z∈D).$

This condition defines a new class of (p,q)-convex functions with respect to symmetric points, denoted by $Cs(p,q)$. For p=1, the class $Cs(q)≡Cs(1,q)$ is a special case of a more general class $Cs(ϕ;p,q)$ where ϕ is as defined in Remark 1.2. The class $Cs(ϕ;1,q)$ was studied in [24]. We also note that, the class

$Cs≡limq→1−Cs1+z1−z;1,q=f∈S:Re2(zf′(z))′(f(z)−f(−z))′>0, z∈D$

was introduced and studied by Das and Singh [10].

### Remark 1.5.

For special values of parameters p and q, we get the following new classes as special cases of Definitions 1.1 and 1.3; for example:

• (1) $STqφ0(z):=STS1,qφ0(z)$,

• (2) $CTqφ0(z):=CS1,qφ0(z)$,

• (3) $STφ0(z):=limq→1−STS1,qφ0(z)$

• (4) $CTφ0(z):=limq→1−CS1,qφ0(z)$.

The sharp bounds of the initial coefficients yield the information regarding the geometric properties like the growth, distortion and covering estimates of the functions. Sharp estimates of the coefficient functional $|a3−μa22|$ for various subclasses of the class $S$ have been computed by many authors, see [12, 16, 18, 19, 25, 28]. The logarithmic coefficients $γn$ of $f∈S$ are defined by the following series expansion:

$logf(z)z=2∑ n=1∞γnzn, z∈D.$

The nth logarithmic coefficient of the Koebe function $k(z)=z/(1−eiθz)2$ is $γn=einθ/n$ for each &#_120579; and for all n ≥ 1. The authors in [11] computed the sharp bound of nth logarithmic coefficient of every univalent function of the type (1.1) by using the work of Baernstein on integral means. In 2018, the researchers in [4, 22] determined the estimates of logarithmic coefficients for certain subclasses of close-to-convex functions. Recently, SokÓŁ and Thomas [29] proved the sharp inequalities for the coefficients of $log(f(z)/z)$ for the subclass of starlike functions associated with Bernoulli lemniscate. Recent details can be seen in [9].

Motivated by the stated research papers, we determine the sharp bounds on (p,q)-Fekete-Szegö inequalities for a function f respectively in the classes $STSp,qφ0(z)$ and $CSp,qφ0(z)$. The estimates of the initial (p,q)-logarithmic coefficients have also been obtained for these classes. In the last section, taking p=1, $0, q-Bieberbach-de-Branges type inequalities are established for these classes.

### 2. (p,q)-Fekete-Szegö Functional Inequalities

In this section, we investigate the behaviour of the Fekete-Szegö functionals defined on $STSp,qφ0(z)$ and $CSp,qφ0(z)$ associated with the Cardioid shaped bounded region given by Figure 1.

### Theorem 2.1.

Let $0.57735≈1/3 and f given by (1.1) belongs to the class $STSp,qφ0(z)$ and µ is any complex number. Then

$|a3−μa22|≤1k([3]p,q−1)max1,([3]p,q −1)μ−2[2]p,q2k[2]p,q2.$

The result is sharp.

In order to prove Theorem 2.1, we need next two results.

### Lemma 2.2.

([20]) If $ψ(z)=1+c1z+c2z2+c3z3+⋯$ is in class $P$ and µ is a complex number, then

$|c2−μc12|≤2max{1,|2μ−1|}.$

The result is sharp for the functions given by

$ψ(z)=1+z21−z2 and ψ(z)=1+z1−z.$

### Lemma 2.3.

([20]) If $ψ(z)=1+c1z+c2z2+c3z3+⋯$ is in class $P$ and µ is a real number, then

$|c2−μc12|≤−4μ+2,if μ≤0,2, if 0≤μ≤1,4μ−2,if μ≥1.$

When $μ<0$ and $μ>1$, equality holds if and only if $ψ(z)=(1+z)/(1−z)$ or one of its rotations. If $0<μ<1$, then equality holds if and only if $ψ(z)=(1+z2)/(1−z2)$ or one of its rotations. If $μ=0$, equality holds if and only if

$ψ(z)=1+λ21+z1−z+1−λ21−z1+z, 0≤λ≤1$

or one of its rotations. If $μ=1$, equality holds if and only if $ψ$ is the reciprocal of one of the functions such that the equality holds in the case $μ=0$.

Also the upper bound in inequality (2.1) is sharp, and it can be improved as follows:

$|c2−μc12|+μ|c1|2≤2 0≤μ≤12,$

and

$|c2−μc12|+(1−μ)|c1|2≤2 12<μ≤1$

where $0<μ<1$.

Proof of Theorem 2.1. Since $f∈STSp,qφ0(z)$, it follows that

$2zDp,qf(z)f(z)−f(−z)=φ0(u(z)),$

where $u:D→D$ is a Schwarz function with $u(0)=0$. Define $ψ:D→ℂ$ by

$ψ(z)=1+u(z)1−u(z)=1+c1z+c2z2+c3z3+⋯,$

or equivalently

$u(z)=ψ(z)−1ψ(z)+1=12(c1z+(c2−c122)z2+14(c13−4c1c2+4c3)z3+⋯).$

Then $ψ$ is analytic in $D$ with $ψ(0)=1$. Since $u:D→D$, the function $ψ$ has positive real part in $D$, and hence $|ci|≤2$ for all $i≥1$. From (1.2), we get

$φ0(u(z))=1+12kc1z+12kc2−c122+12k2c12z2 +z3k2c13−4k2c2c1+4k2c3−4kc13+8kc2c1+2c138k3+⋯.$

Using (2.3) in (2.2), and on comparing both sides we get

$[2]p,qa2=c112k,$ $([3]p,q−1)a3=12kc2−c122+12k2c12,8k3([3]p,q−1)[4]p,qa4=(−k(k−3)+(k2−4k+2)[3]p,q)c13$ $+2k(−3+2k−2(k−2)[3]p,q)c1c2+4k2([3]p,q−1)c3.$

Since $1/3, it follows that $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$. Taking the absolute values of the above two equations and using the fact that $|c1]≤2$ and $|c2−c12/2|≤2$, we obtain

$|a2|≤1k[2]p,q,$

and

$|a3|≤k+2k2([3]p,q−1).$

Using (2.5) and (2.6), we obtain

$a3−μa22=1([3]p,q−1)12kc2−c122+c122k2−μc12[2]p,q24k2 =12k([3]p,q−1)c2−νc12,$

where

$ν=([3]p,q−1)μ−(2−k)[2]p,q22k[2]p,q2.$

In view of Lemma 2.2, we get

$|a3−μa22|≤1k([3]p,q−1)max{1,|2ν−1|},$

where ν is given by (2.8). After substituting the value of |2ν-1|, we get the desired result. The result is sharp for the functions

$2zDp,qf(z)f(z)−f(−z)=φ0(z2) and 2zDp,qf(z)f(z)−f(−z)=φ0(z).$

Putting p=1 and $q→1−$, we get the following new result for the class $STSφ0(z)$:

### Corollary 2.4.

If a function f in the class $STSφ0(z)$ and µ a complex number, then

$|a3−μa22|≤12kmax1,μ−42k.$

The result is sharp.

### Theorem 2.5.

Let $0.57735≈1/3, $f∈STSp,qφ0(z)$ and µ is any real number. If

$σ1=(2−k)[2]p,q2[3]p,q−1, σ2=(2+k)[2]p,q2[3]p,q−1, σ3=2[2]p,q2[3]p,q−1,$

then

$|a3−μa22|≤2[2]p,q2−μ([3]p,q−1)k2([3]p,q−1)[2]p,q2,if μ≤σ1,1k([3]p,q−1), if σ1≤μ≤σ2,([3]p,q−1)μ−2[2]p,q2k2([3]p,q−1)[2]p,q2,if μ≥σ2.$

Further, if $σ1≤μ≤σ3$, then

$|a3−μa22|+([3]p,q−1)μ−(2−k)[2]p,q2)[3]p,q−1|a2|2≤1k([3]p,q−1).$

If $σ3≤μ≤σ2$, then

$|a3−μa22|+((2+k)[2]p,q2−([3]p,q−1)μ)[3]p,q−1|a2|2≤1k([3]p,q−1).$

Sharpness holds for all the inequalities.

Proof. Using equation (2.7), Lemma 2.3 and the fact $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$ for $1/3, we obtain our results. The bounds are sharp as can be seen by defining the following functions for n=2,3 and $0≤λ≤1$.

$2zDp,qFn(z)Fn(z)−Fn(−z)=φ0(zn−1), Fn(0)= F′ n(0)−1=0,$ $2zDp,qGλ(z)Gλ(z)−Gλ(−z)=φ0z(z+λ)λz+1, Gλ(0)= G′λ(0)−1=0,$ $2zDp,qHλ(z)Hλ(z)−Hλ(−z)=φ0−(λz+1)z(z+λ), Hλ(0)= H′λ(0)−1=0.$

It is a routine to verify that these functions belong to $STSp,qφ0(z)$. When $μ<σ1$ or $μ>σ2$, equality holds if and only if f is $F2$ or one of its rotations. When $σ1<μ<σ2$, equality holds if and only if f is $F3$ or one of its rotations. If $μ=σ1$, equality holds if and only if f is $Gλ$ or one of its rotations and if $μ=σ2$, equality holds if and only if f is $Hλ$ or one of its rotations.

Again replacing p by 1 and $q→1−$ in Theorem 2.5, we get the following new result for the class $STSφ0(z)$:

### Corollary 2.6.

Let $f∈STSφ0(z)$ and µ is any real number. If $σ1=2(2−k)$, $σ2=2(2+k)$ and $σ3=4$, then

$|a3−μa22|≤4−μ4k2,if μ≤σ1,12k, if σ1≤μ≤σ2,μ−44k2,if μ≥σ2.$

Further, if $σ1≤μ≤σ3$, then

$|a3−μa22|+(μ−2(2−k))|a2|2≤12k.$

If $σ3≤μ≤σ2$, then

$|a3−μa22|+(2(2+k)−μ)|a2|2≤12k.$

Sharpness holds for all the inequalities.

### Theorem 2.7.

Let $0.57735≈1/3, $f∈CSp,qφ0(z)$ and µ is any complex number. Then

$|a3−μa22|≤1k[3]p,q([3]p,q−1)max{1,|[3]p,q([3]p,q−1)μ−2[2]p,q4k[2]p,q4|}.$

The result is sharp.

Proof. Since $f∈CSp,qφ0(z)$, it follows that

$2Dp,q(zDp,qf(z))Dp,q(f(z)−f(−z))=φ0(u(z)),$

where $u:D→D$ is a Schwarz function with u(0)=0. Define $ψ:D→ℂ$ by

$ψ(z)=1+u(z)1−u(z)=1+c1z+c2z2+...$

or equivalently

$u(z)=ψ(z)−1ψ(z)+1=12(c1z+(c2−c122)z2+14(c13−4c1c2+4c3)z3+...).$

Then ψ is analytic in $D$ with $ψ(0)=1$. Since $u:D→D$, the function ψ has positive real part in $D$, and hence $|ci|≤2$ for $i≥1$. Therefore we have

$φ0(u(z))=1+12kc1z+(12k(c2−c122)+12k2c12)z2+⋯.$

Using (2.9) and (2.10), we get

$[2]p,q2a2=12kc1$ $[3]p,q([3]p,q−1)a3=12k(c2−c122)+12k2c12.$

Since $1/3, it follows that $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$.

Using the method of proof used in Theorem 2.1, we obtain

$a3−μa22=1[3]p,q([3]p,q−1)(12k(c2−c122)+12k2c12)−μc124k2[2]p,q4 =12k[3]p,q([3]p,q−1)(c2−νc12),$

where

$ν=[3]p,q([3]p,q−1)μ−(2−k)[2]p,q42k[2]p,q4.$

Using Lemma 2.2, we get

$|a3−μa22|≤2max{1,|2ν−1|}.$

Substituting the value of $|2ν−1|$ completes the proof. The result is sharp for the functions

Replacing p by 1 and $q→1−$ in Theorem 2.7, we get the following new result for the class $CSφ0(z)$.

### Corollary 2.8.

If the function f is in the class $CSφ0(z)$ and $μ$ a complex number, then

$|a3−μa22|≤16kmax1,3μ−168k.$

The result is sharp.

### Theorem 2.9.

Let $0.57735≈1/3, $f∈CSp,qφ0(z)$ and µ is any real number and let

$δ1=(2−k)[2]p,q4[3]p,q([3]p,q−1), δ2=(2+k)[2]p,q4[3]p,q([3]p,q−1) and δ3=2[2]p,q4[3]p,q([3]p,q−1).$

Then

$|a3−μa22|≤2[2]p,q4−μ[3]p,q([3]p,q−1)k2[3]p,q([3]p,q−1)[2]p,q4,μ≤δ11k[3]p,q([3]p,q−1),δ1≤μ≤δ2[3]p,q([3]p,q−1)μ−2[2]p,q4k2[3]p,q([3]p,q−1)[2]p,q4,μ≥δ2$

Moreover, if $δ1≤μ≤δ3$, then

$|a3−μa22|+[3]p,q([3]p,q−1)μ−(2−k)[2]p,q4[3]p,q([3]p,q−1)|a2|2≤1k[3]p,q([3]p,q−1),$

and if $δ3≤μ≤δ2$, then

$|a3−μa22|+(2+k)[2]p,q4−[3]p,q([3]p,q−1)μ[3]p,q([3]p,q−1)|a2|2≤1k[3]p,q([3]p,q−1).$

These results are sharp.

Proof. Using Lemma 2.3, (2.11), and the fact $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$ for $1/3, (2.12), we get the following results.

If $μ≤δ1$, then

$|a3−μa22|≤2[2]p,q4−μ[3]p,q([3]p,q−1)k2[3]p,q([3]p,q−1)[2]p,q4.$

If $δ1≤μ≤δ2$, then

$|a3−μa22|≤1k[3]p,q([3]p,q−1).$

If $μ≥δ2$, then

$|a3−μa22|≤[3]p,q([3]p,q−1)μ−2[2]p,q4k2[3]p,q([3]p,q−1)[2]p,q4.$

Since $1/3, it follows that $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$. On the other hand, using (2.11) and (2.12) for the values $δ1≤μ≤δ3$ we have

$|a3−μa22|+(μ−δ1)|a2|2=12k[3]p,q([3]p,q−1)|c2−νc12| +(μ−(2−k)[2]p,q4[3]p,q([3]p,q−1))c124k2[2]p,q4 =1k[3]p,q([3]p,q−1)(12|c2−νc12|+νc12) ≤1k[3]p,q([3]p,q−1).$

Similarly, if $δ3≤μ≤δ2$ we have

$|a3−μa22|+(δ2−μ)|a2|2≤1k[3]p,q([3]p,q−1).$

Similar to the results in previous corollaries, putting p=1 and $q→1−$, the Fekete-Szegö functionals can be obtained for the class $CSφ0(z)$.

In this section, we obtain the estimates of the initial logarithmic coefficients of the functions f defined by (1.1) in $STSp,q(φ0)$ and $CSp,qφ0(z)$.

### Theorem 3.1.

Let $0.57735≈1/3 and f given by (1.1) belongs to class $STSp,q(φ0)$. Then the estimates for initial logarithmic coefficients are given by

$|γ1|≤12k.1[2]p,q,$ $|γ2|≤14k([3]p,q−1)max1,([3]p,q −1)−4[2]p,q22k[2]p,q2,$

and

$|γ3|≤12k[4]p,q.H(ξ1;ξ2),$

where

$ξ1=([2]p,q(3−2k+2(−2+k)[3]p,q)+[4]p,q)2k[2]p,q(1−[3]p,q),$ $ξ2=3[2]p,q3(k(3+k(−1+[3]p,q)−4[3]p,q)+2[3]p,q)+(−1−3k[2]p,q2+[3]p,q)[4]p,q12k2[2]p,q3([3]p,q−1)$

and $H(ξ1;ξ2)$ is as in Lemma 3.2.

In order to prove Theorem 3.1, we need following result due to Prokhorov and Szynal.

### Lemma 3.2.

([23]) Let $w(z)=1+∑ n=1∞cnzn$ be a Schwarz function. Then for any real numbers q1 and q2, the following sharp inequality holds:

$|c3+q1c1c2+q2c13|≤H(q1;q2),$

where

and for $k=1,2,⋯12$, the sets Dk are defined as

$D1=(q1,q2):|q1|≤12,|q2|≤1,D2=(q1,q2):12≤|q1|≤2,427(|q1|+1)3−(|q1|+1)≤|q2|≤1,D3=(q1,q2):|q1|≤12,|q2|≤−1,D4=(q1,q2):|q1|≥12,|q2|≤−23(|q1|+1),D5=(q1,q2):|q1|≤2,|q2|≥1,D6=(q1,q2):2≤|q1|≤4,|q2|≥112(|q1|2+8),D7=(q1,q2):|q1|≥4,|q2|≥23(|q1|−1),D8=(q1,q2):12≤|q1|≤2,−23(|q1|+1)≤|q2|≤427(|q1|+1)3−(|q1|+1),D9=(q1,q2):|q1|≥2,−23(|q1|+1)≤|q2|≤2|q1|(|q1|+1)q12+2|q1|+4,D10=(q1,q2):2≤|q1|≤4,2|q1|(|q1|+1)q12+2|q1|+4≤|q2|≤112(|q1|2+8),D11=(q1,q2):|q1|≥4,2|q1|(|q1|+1)q12+2|q1|+4≤|q2|≤2|q1|(|q1|−1)q12−2|q1|+4,D12=(q1,q2):|q1|≥4,2|q1|(|q1|−1)q12−2|q1|+4≤|q2|≤23(|q1|−1).$

Proof of Theorem 3.1. In view of series expansion (1.10), it follows that

$γ1=a2/2,$ $γ2=(a3−12a22)/2,$ $γ3=(a4−a2a3+13a23).$

By substituting the value of a2 from (2.4) in (3.3), we get the desired bound of $γ1$. Since $1/3, it follows that $[3]p,q−1=p2+pq+q2−1≥3q2−1>0$. Further, we substitute the value of a2 and a3 from (2.4) and (2.5) in (3.4), then

$|γ2|=14k([3]p,q−1)|c2−ζc12|,$

where

$ζ=([3]p,q−1)−2(2−k)[2]p,q24k[2]p,q2.$

Using Lemma 2.3, we get the required bound for γ2. On putting the values of a2, a3 and a4 from (2.4), (2.5) and (2.6) in (3.5), we have

$γ3=G(c1,c2,c3)24k3[2]p,q3([3]p,q−1)[4]p,q$

where

$G(c1,c2,c3)=12c3k2[2]p,q3([3]p,q−1)−6k[2]p,q2([2]p,q(3−2k+2(−2+k)[3]p,q) +[4]p,q)c1c2+(3[2]p,q3(k(3+k(−1+[3]p,q)−4[3]p,q)+2[3]p,q) +(−1−3k[2]p,q2+[3]p,q)[4]p,q)c13.$

On simplifying,

$|γ3|≤12k[4]p,q|c3+ξ1c1c2+ξ2c13|,$

where $ξ1$ and $ξ2$ are given by (3.1) and (3.2) respectively. By making use of Lemma 3.2 in inequality (3.6), we get the desired result.

The proof of next theorem is similar to the proof of previous theorem and hence it is omitted.

### Theorem 3.3.

Let $0.57735≈1/3 and $f∈A$ be a function in $CSp,qφ0(z)$. Then the estimates for initial logarithmic coefficients are given by

$|γ1|≤12k.1[2]p,q2,$ $|γ2|≤14k[3]p,q([3]p,q−1)max1,[3]p,q ([3]p,q −1)−4[2]p,q42k[2]p,q4,$

and

$|γ3|≤ζ3H(ζ2/ζ3;ζ1/ζ3)$

where

$ζ1=3[2]p,q6[3]p,q(k(3−k)+(k2−4k+2)[3]p,q)+3(k−2)[2]p,q4[4]p,q2+[3]p,q([3]p,q−1)[4]p,q248k3[2]p,q6[3]p,q[4]p,q2([3]p,q−1)ζ2=6k[2]p,q6[3]p,q(−3+2k−2(k−2)[3]p,q)−6k[2]p,q4[4]p,q248k3[2]p,q6[3]p,q[4]p,q2([3]p,q−1)ζ3=14k[4]p,q2$

and $H(ζ2/ζ3;ζ1/ζ3)$ is defined as in Lemma 3.2.

### 4. q-Bieberbach-de-Branges Type Coefficient Inequalities

In view of the work done in [6, 27], we investigate q-Bieberbach-de-Branges type coefficient inequalities for functions belonging to the classes $STSqφ0(z)$ and $CSqφ0(z)$.

### Theorem 4.1.

If a function f of the form (1.1) is in $STSqφ0(z)$, then for $n≥1$

$|a2n|≤k+1k2n[2n]q∏ j=1 n−1k+1+k([2j+1]q−1)k([2j+1]q−1),$ $|a2n+1|≤k+1k2n+1([2n+1]q−1)∏ j=1 n−1k+1+k([2j+1]q−1)k([2j+1]q−1),$

where $k=2+1$.

We need the following result for the proof of Theorem 4.1

### Lemma 4.2.

Let $ψ(z)=1+∑ k=1∞cnzn∈P$. If $ψ(z)≺φ0(z)=1+z(k+z)/k(k−z)$, $k=2+1$, then we have

$|cn|≤k+1kn+1$

for all $n≥1$.

Proof. Since $ψ(z)≺1+z(k+z)/k(k−z)$, we can write

$ψ(z)=k2+(w(z))2k2−kw(z),$

where w is Schwarz function with w(0)=0 and $|w(z)|<1.$ Using (4.4), we obtain the following

$k2(ψ(z)−1)=(kψ(z)+w(z))w(z),k2(c1z+c2z2+...)=[k(1+c1z+c2z2+...)+w(z)]w(z),k2∑n=1∞cnzn=[k+k∑n=1∞cnzn+w(z)]w(z),k2∑n=1scnzn+k2∑n=s+1∞dnzn=[k+k∑n=1s−1cnzn+k∑n=s∞cnzn+w(z)]w(z).$

Using Parseval identity, and |z|=r, we get

$k4∑n=1s|cn|2r2n≤k4∑n=1s|cn|2r2n+k4∑n=s+1∞|dn|2r2n ≤12π∫02π|k+k∑n=1s−1cnrn+w(z)|2|w(z)|2dz ≤12π∫02π|k+k∑n=1s−1cnrn+w(z)|2dz ≤(k+1)2+k2∑n=1s−1|cn|2|r|2n.$

If $r→1−$, then

$k4∑ n=1s|cn|2≤(k+1)2+k2∑ n=1 s−1|cn|2.$

This gives desired result for all $n≥1.$

Proof of Theorem 4.1. In view of subordination, a function $f∈STq(φ0)$ if and only if

$2zDqf(z)f(z)−f(−z)=1+w(z)(k+w(z))k(k−w(z)),$

where w is a Schwarz function with w(0)=0 and $|w(z)|<1.$ Letting

$1+w(z)(k+w(z))k(k−w(z))=ψ(z)=1+∑ k=1∞cnzn,$

we have

$2zDqf(z)=(f(z)−f(−z))ψ(z),$

where $ψ(z)≺φ0(z)=1+z(k+z)/k(k−z)$. This gives

$z+[2]qa2z2+[3]qa3z3+...=(z+a3z3+a5z5+...)(1+c1z+c2z2+...).$

Equating both sides of the equality (4.6), we get

$[2]qa2=c1,$ $([3]q−1)a3=c2,$ $[4]qa4=c3+a3c1,$ $([5]q−1)a5=c4+a3c2,$

and for n'th term, we get

$[2n]qa2n=c2n−1+a3c2n−3+...+a2n−1c1,$ $([2n+1]q−1)a2n+1=c2n+a3c2n−2+...+a2n−1c2.$

In view of Lemma 4.2 and using (4.7), (4.8), (4.9) and (4.10) we get the following coefficient inequalities.

$|a2|≤k+1k2[2]q, |a3|≤k+1k3([3]q−1),$ $|a4|≤k+1k4[4]q(k([3]q−1)+k+1k([3]q−1)),$ $|a5|≤k+1k5([5]q−1)(k([3]q−1)+k+1k([3]q−1)).$

These bounds show that (4.1) and (4.2) are valid for n=1 and n=2.

We now prove (4.1) and (4.2) by using induction. Taking into account Lemma 4.2, (4.11) and (4.12) we get

$|a2n|≤k+1k2n[2n]q(1+∑ s=1 n−11k−2s|a2s+1|)$

and

$|a2n+1|≤k+1k2n+1([2n+1]q−1)(1+∑ s=1 n−11k−2s|a2s+1|).$

Now, suppose that (4.1) and (4.2) hold for s ∈ {3,4,...,n-1}. In order to prove (4.1), we will use (4.13) and get

$|a2n|≤k+1k2n[2n]q(1+∑ s=1 n−1k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1)).$

In order to complete the proof, it is sufficient to show that

$k+1k2m[2m]q∏ j=1 m−1k+1+k([2j+1]q−1)k([2j+1]q−1)=k+1k2m[2m]q(1+∑ s=1 m−1k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1))$

for $m∈{3,4,...,n}$. We note that (4.15) is valid for m=3. Assume that (4.15) holds for $4≤m≤n−1$. Then we obtain

$k+1k2n[2n]q(1+∑ s=1 n−1k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1)) =k+1k2n[2n]q(1+∑ s=1 n−2k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1))+ k+1k2n[2n]q.k+1k([2n−1]q−1)∏ j=1 n−2k+1+k([2j+1]q−1)k([2j+1]q−1) =k+1k2n[2n]q.k2(n−1)[2(n−1)]qk2(n−1)[2(n−1)]q(1+∑ s=1 n−2k+1k([2s+1]q−1) ∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1))+ k+1k2n[2n]qk+1k([2n−1]q−1)∏ j=1 n−2k+1+k([2j+1]q−1)k([2j+1]q−1) =k+1k2n[2n]q∏ j=1 n−1k+1+k([2j+1]q−1)k([2j+1]q−1).$

Therefore, (4.15) is valid for m=n. This completes the proof of (4.1). Similarly, using (4.14) we obtain (4.2).

### Theorem 4.3.

If a function f of the form (1.1) is in the class $CSqφ0(z)$, then for $n≥1$

$|a2n|≤k+1k2n[2n]q2∏ j=1 n−1k+1+k([2j+1]q−1)k([2j+1]q−1),$ $|a2n+1|≤k+1k2n+1[2n+1]q([2n+1]q−1)∏ j=1 n−1k+1+k([2j+1]q−1)k([2j+1]q−1).$

Proof. In view of subordination, a function $f∈CSq(φ0)$ if and only if

$2Dq(zDqf(z))Dq(f(z)−f(−z))=1+w(z)(k+w(z))k(k−w(z)),$

where w is a Schwarz function with w(0)=0 and $|w(z)|<1.$ Using equation (4.5), we have

$2zDq(zDqf(z))=zDq(f(z)−f(−z))ψ(z),$

where $ψ(z)≺φ0(z)=1+z(k+z)/k(k−z)$. This gives

$z+[2]q2a2z2+[3]q2a3z3+...=(z+[3]qa3z3+[5]qa5z5+...)(1+c1z+c2z2+...).$

Equating both sides, we get

$[2]q2a2=c1,$ $[3]q([3]q−1)a3=c2,$ $[4]q2a4=c3+[3]qa3c1,$ $[5]q([5]q−1)a5=c4+[3]qa3c2,$

and for n'th term we get

$[2n]q2a2n=c2n−1+[3]qa3c2n−3+...+[2n−1]qa2n−1c1,$ $[2n+1]q([2n+1]q−1)a2n+1=c2n+[3]qa3c2n−2+...+[2n−1]qa2n−1c2.$

Using Lemma 4.2 and (4.18), (4.19), (4.20) and (4.21) we get the following coefficient inequalities.

$|a2|≤k+1k2[2]q2, |a3|≤k+1k3[3]q([3]q−1),$ $|a4|≤k+1k4[4]q2(k([3]q−1)+k+1k([3]q−1)),$ $|a5|≤k+1k5[5]q([5]q−1)(k([3]q−1)+k+1k([3]q−1)).$

These bounds show that (4.16) and (4.17) are valid for n=1 and n=2.

We now prove (4.16) and (4.17) by using induction. Using Lemma 4.2 and equations (4.22) and (4.23) we get

$|a2n|≤k+1k2n[2n]q2(1+∑ s=1 n−11k−2s[2s+1]q|a2s+1|)$

and

$|a2n+1|≤k+1k2n+1[2n+1]q([2n+1]q−1)(1+∑ s=1 n−11k−2s[2s+1]q|a2s+1|).$

Now, suppose that (4.16) and (4.17) hold for $s∈{3,4,...,n−1}$. In order to prove (4.16), we will use (4.24) and get

$|a2n|≤k+1k2n[2n]q2(1+∑ s=1 n−1k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]p,q−1)k([2j+1]q−1)).$

In order to complete the proof, it is sufficient to show that

$k+1k2m[2m]q2∏ j=1 m−1k+1+k([2j+1]q−1)k([2j+1]q−1)=k+1k2m[2m]q2(1+∑ s=1 m−1k+1k([2s+1]q−1)∏ j=1 s−1k+1+k([2j+1]q−1)k([2j+1]q−1))$

for $m∈{3,4,...,n}$. We note that (4.26) is valid for m=3. Assume that (4.26) holds for $4≤m≤n−1$. Then using the same method given in Theorem 4.1, we prove that (4.26) is valid for m=n. This completes the proof of (4.16). Similarly, using (4.25) we obtain (4.17).

In Remark 1.2 and Remark 1.4, we defined two new classes $Ss*(ϕ;p,q)$ and $Cs(ϕ;p,q)$ where ϕ is an analytic function with positive real part in $D$ with ϕ(0)=0 and $ϕ′(0)>0$. We omitted analogous results and proofs of these two new classes because techniques are similar to the corresponding techniques used for $STSp,qφ0(z)$ and $CTSp,qφ0(z)$ in this paper. However, results for all these classes are expected to be different.

All the authors are thankful to the referees for giving helpful comments/suggestions.

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