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

(1.1)$$f(z)=z+{\displaystyle \sum _{n=2}^{\infty }{a}_n}{z}^n(z\in \mathbb{D}).$$

Let $\mathcal{S}$ be the subclass of $\mathcal{A}$ containing all the univalent functions in $\mathbb{D}$. A function $f\in \mathcal{A}$ is starlike if and only if RE $\left(z{f}^{\prime }(z)/f(z)\right)>0$ and convex if and only if RE$\left(1+z{f}^{″}(z)/f^{\prime }(z)\right)>0$ for all $z\in \mathbb{D}$. We denote these two classes, respectively, by ${\mathcal{S}}^{*}$ and $\mathcal{K}$. Closely related with these two classes is the class $\mathcal{P}$ of all functions $\psi$ which are analytic and have positive real part in $\mathbb{D}$ with $\psi (0)=1$. An analytic function f is subordinate to an analytic function g, denoted by $f\prec g$, if there is an analytic function w defined on $\mathbb{D}$ 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 ${\mathcal{S}}_R^{*}={\mathcal{S}}^{*}({\phi }_0)$, where ${\phi }_0$ is a rational function given by

(1.2)$${\phi }_0(z):=1+\frac{z}{k}\left(\frac{k+z}{k-z}\right)=1+\frac{1}{k}z+\frac{2}{k^2}{z}^2+\frac{2}{k^3}{z}^3+\cdots ,k=\sqrt{2}+1.$$

Note that the image of the open unit disk under the rational function ${\phi }_0(z)$ 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 $0<q\le p\le 1$. The (p,q)-bracket or twin basic number ${[n]}_{p,q}$ is defined by

Figure 1. Image of unit disk $\mathbb{D}$ under the function ${\phi }_0$

$${[n]}_{p,q}=\left\{\begin{array}{cc}\frac{p^n-q^n}{p-q},& \text{if}q\ne p,\\ n{p}^{n-1},& \text{if}q=p.\end{array}\right.$$

For $0<q<1$, q-bracket [n]_q for $n=0,1,2,\dots$ is given by ${[n]}_q={[n]}_{1,q}$. The (p,q)-derivative of a function f is defined by

$$D_{p,q}f(z)=\left\{\begin{array}{cc}\frac{f(pz)-f(qz)}{(p-q)z},& \text{if}p\ne q,z\ne 0,\\ 1,& \text{if}p\ne q,z=0,\\ f^{\prime }(z),& \text{if}p=q.\end{array}\right.$$

In particular, $D_{p,q}{z}^n={[n]}_{p,q}{z}^{n-1}$. For a function $f\in \mathcal{A}$, the (p,q)-derivative operator is given by ${\text{D}}_{p,q}f(z)=1+{\displaystyle {\sum }_{n=2}^{\infty }{[n]}_{p,q}{a}_n}{z}^{n-1}$. For definitions and properties of (p,q)-calculus, one may refer to [8]. The (1,q) derivative operator D_1,q, denoted by D_q is known as the q-derivative operator. Jackson in 1909 and 1910 [14, 15], initiated the study of q-operator D_q defined by

$$D_{q}f(z)=\left\{\begin{array}{cc}\frac{f(z)-f(qz)}{(1-q)z},& \text{if}0<q<1,z\ne 0,\\ 1,& \text{if}q\to 1^{-},z=0,\\ f^{\prime }(z),& \text{if}q\to 1^{-},z\ne 0.\end{array}\right.$$

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 $f,g\in \mathcal{A}$ with

$$f(z)={\displaystyle \sum _{n=1}^{\infty }{a}_n}{z}^n\text{and}g(z)={\displaystyle \sum _{n=1}^{\infty }{b}_n}{z}^n,$$

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

$$f(z)\ast g(z)=(f\ast g)(z)=z+{\displaystyle \sum _{n=2}^{\infty }{a}_n}{b}_n{z}^n.$$

If $g(z)=z/(1-z)$, then obviously $f\ast g=f$ for all $f\in \mathcal{A}$. Again, if $g(z)=z/{(1-z)}^2,$ then it is straightforward to notice that $f\ast g=z{f}^{\prime }$ for all $f\in \mathcal{A}$.

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 ${\phi }_0$.

Definition 1.1.

A function $f\in \mathcal{S}$ is said to be in class $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ if it satisfies the following subordination condition:

(1.3)$$\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}\prec {\phi }_0(z),(z\in \mathbb{D}).$$

where ${\phi }_0$ is a rational function defined by (1.2).

The functions in the class $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ may be called as (p,q)-starlike functions with respect to symmetric points associated with the rational function ${\phi }_0$. Using the concept of the subordination, we say that there exists a function w analytic in $\mathbb{D}withw(0)=0,|w(z)|<1$ such that

$$\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}=1+\frac{w(z)(k+w(z))}{k(k-w(z))},$$

which is equivalent to

$$\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}\ne \frac{k^2+e^{2i\theta }}{k^2-k{e}^{i\theta }},(z\in \mathbb{D};\theta \in [0,2\pi ])$$

or

(1.4)$$2(k^2-k{e}^{i\theta })z{D}_{p,q}f(z)-(k^2+e^{2i\theta })(f(z)-f(-z))\ne 0$$

In view of convolution, we obtain

(1.5)$$f(z)-f(-z)=f(z)\ast \frac{z}{1-z}-f(z)\ast \frac{-z}{1-(-z)}=2f(z)\ast \frac{z}{(1-z)(1+z)}$$

and

(1.6)$$z{D}_{p,q}f(z)=f(z)\ast \frac{z}{(1-pz)(1-qz)}.$$

On substituting (1.5) and (1.6) into (1.4), we conclude that the function f is in the class $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ if and only if

(1.7)$$\frac{1}{z}[f(z)\ast (\frac{2(k^2-k{e}^{i\theta })z}{(1-pz)(1-qz)}-\frac{2(k^2+e^{2i\theta })z}{(1-z)(1+z)})]\ne 0.$$

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 ${\phi }_0$ in Definition 1.1 by the function (1+z)/(1-z), then the subordination condition (1.3) reduces to the inequality

$$\mathrm{Re}\left(\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}\right)>0,(z\in \mathbb{D});$$

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

$${\mathcal{S}}_s^{*}\equiv \underset{q\to 1^{-}}{\lim }{\mathcal{S}}_s^{*}\left(\frac{1+z}{1-z};1,q\right)=\left\{f\in \mathcal{S}:\mathrm{Re}\left(\frac{2z{f}^{\prime }(z)}{f(z)-f(-z)}\right)>0,z\in \mathbb{D}\right\}$$

was introduced and studied by Sakaguchi [26].

Definition 1.3.

A function $f\in \mathcal{S}$ is said to be in class $\mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ if the following subordination condition holds:

(1.8)$$\frac{2D_{p,q}(z{D}_{p,q}f(z))}{D_{p,q}(f(z)-f(-z))}\prec {\phi }_0(z),(z\in \mathbb{D}).$$

where ${\phi }_0$ is a rational function defined by (1.2).

The functions in the class $\mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ may be called as (p,q)-convex functions with respect to symmetric points associated with the rational function ${\phi }_0$. Applying the method used in proving (1.7), we find that the function f is in the class $\mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ if and only if

(1.9)$$\frac{1}{z}[f(z)\ast (\frac{2(k^2-k{e}^{i\theta })(z+pq{z}^2)}{(1-p^{2}z)(1-q^{2}z)(1-pqz)}-\frac{2(k^2+e^{2i\theta })(z+pq{z}^3)}{(1-p^2{z}^2)(1+q^2{z}^2)})]\ne 0$$

Remark 1.4.

If we replace the rational function ${\phi }_0$ in Definition 1.3 by the function (1+z)/(1-z), then the subordination condition (1.8) reduces to the inequality

$$\mathrm{Re}\left(\frac{2D_{p,q}(z{D}_{p,q}f(z))}{D_{p,q}(f(z)-f(-z))}\right)>0,(z\in \mathbb{D}).$$

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

$${\mathcal{C}}_s\equiv \underset{q\to 1^{-}}{\lim }{\mathcal{C}}_s\left(\frac{1+z}{1-z};1,q\right)=\left\{f\in \mathcal{S}:\text{Re}\left(\frac{2{(z{f}^{\prime }(z))}^{\prime }}{{(f(z)-f(-z))}^{\prime }}\right)>0,z\in \mathbb{D}\right\}$$

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) $\mathcal{S}{\mathcal{T}}_q{\phi }_0(z):=\mathcal{S}\mathcal{T}{\mathcal{S}}_{1,q}{\phi }_0(z)$,

• (2) $\mathcal{C}{\mathcal{T}}_q{\phi }_0(z):=\mathcal{C}{\mathcal{S}}_{1,q}{\phi }_0(z)$,

• (3) $\mathcal{S}\mathcal{T}{\phi }_0(z):=\underset{q\to 1^{-}}{\lim }\mathcal{S}\mathcal{T}{\mathcal{S}}_{1,q}{\phi }_0(z)$

• (4) $\mathcal{C}\mathcal{T}{\phi }_0(z):=\underset{q\to 1^{-}}{\lim }\mathcal{C}{\mathcal{S}}_{1,q}{\phi }_0(z)$.

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 $|a_3-\mu a_2^2|$ for various subclasses of the class $\mathcal{S}$ have been computed by many authors, see [12, 16, 18, 19, 25, 28]. The logarithmic coefficients ${\gamma }_n$ of $f\in \mathcal{S}$ are defined by the following series expansion:

(1.10)$$\log \frac{f(z)}{z}=2{\displaystyle \sum _{n=1}^{\infty }{\gamma }_n}{z}^n,z\in \mathbb{D}.$$

The n^th logarithmic coefficient of the Koebe function $k(z)=z/{(1-e^{i\theta }z)}^2$ is ${\gamma }_n=e^{in\theta }/n$ for each &#_120579; and for all n ≥ 1. The authors in [11] computed the sharp bound of n^th 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 $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and $\mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$. The estimates of the initial (p,q)-logarithmic coefficients have also been obtained for these classes. In the last section, taking p=1, $0<q<1$, q-Bieberbach-de-Branges type inequalities are established for these classes.

In this section, we investigate the behaviour of the Fekete-Szegö functionals defined on $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and $\mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ associated with the Cardioid shaped bounded region given by Figure 1.

Theorem 2.1.

Let $0.57735\approx 1/\sqrt{3}<q\le p\le 1$ and f given by (1.1) belongs to the class $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and µ is any complex number. Then

$$|a_3-\mu a_2^2|\le \frac{1}{k({[3]}_{p,q}-1)}\max \left\{1,\left|\frac{({[3]}_{p,q}-1)\mu -2{[2]}_{p,q}^2}{k{[2]}_{p,q}^2}\right|\right\}.$$

The result is sharp.

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

Lemma 2.2.

([20]) If $\psi (z)=1+c_{1}z+c_2{z}^2+c_3{z}^3+\cdots$ is in class $\mathcal{P}$ and µ is a complex number, then

$$|c_2-\mu c_1^2|\le 2\max \{1,|2\mu -1|\}.$$

The result is sharp for the functions given by

$$\psi (z)=\frac{1+z^2}{1-z^2}\text{and}\psi (z)=\frac{1+z}{1-z}.$$

Lemma 2.3.

([20]) If $\psi (z)=1+c_{1}z+c_2{z}^2+c_3{z}^3+\cdots$ is in class $\mathcal{P}$ and µ is a real number, then

(2.1)$$|c_2-\mu c_1^2|\le \left\{\begin{array}{cc}-4\mu +2,& \text{if}\mu \le 0,\\ 2,& \text{if}0\le \mu \le 1,\\ 4\mu -2,& \text{if}\mu \ge 1.\end{array}\right.$$

When $\mu <0$ and $\mu >1$, equality holds if and only if $\psi (z)=(1+z)/(1-z)$ or one of its rotations. If $0<\mu <1$, then equality holds if and only if $\psi (z)=(1+z^2)/(1-z^2)$ or one of its rotations. If $\mu =0$, equality holds if and only if

$$\psi (z)=\frac{1+\lambda }{2}\left(\frac{1+z}{1-z}\right)+\frac{1-\lambda }{2}\left(\frac{1-z}{1+z}\right),0\le \lambda \le 1$$

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

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

$$|c_2-\mu c_1^2|+\mu |c_1{|}^2\le 2\left(0\le \mu \le \frac{1}{2}\right),$$

and

$$|c_2-\mu c_1^2|+(1-\mu )|c_1{|}^2\le 2\left(\frac{1}{2}<\mu \le 1\right)$$

where $0<\mu <1$.

Proof of Theorem 2.1. Since $f\in \mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$, it follows that

(2.2)$$\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}={\phi }_0(u(z)),$$

where $u:\mathbb{D}\to \mathbb{D}$ is a Schwarz function with $u(0)=0$. Define $\psi :\mathbb{D}\to ℂ$ by

$$\psi (z)=\frac{1+u(z)}{1-u(z)}=1+c_{1}z+c_2{z}^2+c_3{z}^3+\cdots ,$$

or equivalently

$$u(z)=\frac{\psi (z)-1}{\psi (z)+1}=\frac{1}{2}(c_{1}z+(c_2-\frac{c_1^2}{2})z^2+\frac{1}{4}(c_1^3-4c_1{c}_2+4c_3)z^3+\cdots ).$$

Then $\psi$ is analytic in $\mathbb{D}$ with $\psi (0)=1$. Since $u:\mathbb{D}\to \mathbb{D}$, the function $\psi$ has positive real part in $\mathbb{D}$, and hence $|c_i|\le 2$ for all $i\ge 1$. From (1.2), we get

(2.3)$$\begin{array}{l}{\phi }_0(u(z))=1+\frac{1}{2k}{c}_{1}z+\left(\frac{1}{2k}\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{2k^2}{c}_1^2\right)z^2\\ +\frac{z^3\left(k^2{c}_1^3-4k^2{c}_2{c}_1+4k^2{c}_3-4k{c}_1^3+8k{c}_2{c}_1+2c_1^3\right)}{8k^3}+\cdots .\end{array}$$

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

(2.4)$${[2]}_{p,q}{a}_2=c_1\frac{1}{2k},$$ (2.5)$$\begin{array}{l}({[3]}_{p,q}-1)a_3=\frac{1}{2k}\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{2k^2}{c}_1^2,\\ 8k^3({[3]}_{p,q}-1){[4]}_{p,q}{a}_4=(-k(k-3)+(k^2-4k+2){[3]}_{p,q})c_1^3\end{array}$$ (2.6)$$+2k(-3+2k-2(k-2){[3]}_{p,q})c_1{c}_2+4k^2({[3]}_{p,q}-1)c_3.$$

Since $1/\sqrt{3}<q\le p\le 1$, it follows that ${[3]}_{p,q}-1=p^2+pq+q^2-1\ge 3q^2-1>0$. Taking the absolute values of the above two equations and using the fact that $|c_1]\le 2$ and $|c_2-c_1^2/2|\le 2$, we obtain

$$|a_2|\le \frac{1}{k{[2]}_{p,q}},$$

and

$$|a_3|\le \frac{k+2}{k^2({[3]}_{p,q}-1)}.$$

Using (2.5) and (2.6), we obtain

(2.7)$$\begin{array}{l}{a}_3-\mu a_2^2=\frac{1}{({[3]}_{p,q}-1)}\left(\frac{1}{2k}\left(c_2-\frac{c_1^2}{2}\right)+\frac{c_1^2}{2k^2}\right)-\mu \frac{c_1^2}{{[2]}_{p,q}^{2}4k^2}\\ =\frac{1}{2k({[3]}_{p,q}-1)}\left(c_2-\nu c_1^2\right),\end{array}$$

where

(2.8)$$\nu =\frac{({[3]}_{p,q}-1)\mu -(2-k){[2]}_{p,q}^2}{2k{[2]}_{p,q}^2}.$$

In view of Lemma 2.2, we get

$$|a_3-\mu a_2^2|\le \frac{1}{k({[3]}_{p,q}-1)}\max \{1,|2\nu -1|\},$$

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

$$\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}={\phi }_0(z^2)\text{and}\frac{2z{D}_{p,q}f(z)}{f(z)-f(-z)}={\phi }_0(z).$$

Putting p=1 and $q\to 1^{-}$, we get the following new result for the class $\mathcal{S}\mathcal{T}\mathcal{S}{\phi }_0(z)$:

Corollary 2.4.

If a function f in the class $\mathcal{S}\mathcal{T}\mathcal{S}{\phi }_0(z)$ and µ a complex number, then

$$|a_3-\mu a_2^2|\le \frac{1}{2k}\max \left\{1,\left|\frac{\mu -4}{2k}\right|\right\}.$$

The result is sharp.

Theorem 2.5.

Let $0.57735\approx 1/\sqrt{3}<q\le p\le 1$, $f\in \mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and µ is any real number. If

$${\sigma }_1=\frac{(2-k){[2]}_{p,q}^2}{{[3]}_{p,q}-1},{\sigma }_2=\frac{(2+k){[2]}_{p,q}^2}{{[3]}_{p,q}-1},{\sigma }_3=\frac{2{[2]}_{p,q}^2}{{[3]}_{p,q}-1},$$

then

$$|a_3-\mu a_2^2|\le \left\{\begin{array}{cc}\frac{2{[2]}_{p,q}^2-\mu ({[3]}_{p,q}-1)}{k^2({[3]}_{p,q}-1){[2]}_{p,q}^2},& \text{if}\mu \le {\sigma }_1,\\ \frac{1}{k({[3]}_{p,q}-1)},& \text{if}{\sigma }_1\le \mu \le {\sigma }_2,\\ \frac{({[3]}_{p,q}-1)\mu -2{[2]}_{p,q}^2}{k^2({[3]}_{p,q}-1){[2]}_{p,q}^2},& \text{if}\mu \ge {\sigma }_2.\end{array}\right.$$

Further, if ${\sigma }_1\le \mu \le {\sigma }_3$, then

$$|a_3-\mu a_2^2|+\frac{({[3]}_{p,q}-1)\mu -(2-k){[2]}_{p,q}^2)}{{[3]}_{p,q}-1}|a_2{|}^2\le \frac{1}{k({[3]}_{p,q}-1)}.$$

If ${\sigma }_3\le \mu \le {\sigma }_2$, then

$$|a_3-\mu a_2^2|+\frac{((2+k){[2]}_{p,q}^2-({[3]}_{p,q}-1)\mu )}{{[3]}_{p,q}-1}|a_2{|}^2\le \frac{1}{k({[3]}_{p,q}-1)}.$$

Sharpness holds for all the inequalities.

Proof. Using equation (2.7), Lemma 2.3 and the fact ${[3]}_{p,q}-1=p^2+pq+q^2-1\ge 3q^2-1>0$ for $1/\sqrt{3}<q\le p\le 1$, we obtain our results. The bounds are sharp as can be seen by defining the following functions for n=2,3 and $0\le \lambda \le 1$.

$$\frac{2z{D}_{p,q}{\mathcal{F}}_n(z)}{{\mathcal{F}}_n(z)-{\mathcal{F}}_n(-z)}={\phi }_0(z^{n-1}),{\mathcal{F}}_n(0)={{\mathcal{F}}^{\prime }}_n(0)-1=0,$$ $$\frac{2z{D}_{p,q}{\mathcal{G}}_{\lambda }(z)}{{\mathcal{G}}_{\lambda }(z)-{\mathcal{G}}_{\lambda }(-z)}={\phi }_0\left(\frac{z(z+\lambda )}{\lambda z+1}\right),{\mathcal{G}}_{\lambda }(0)={{\mathcal{G}}^{\prime }}_{\lambda }(0)-1=0,$$ $$\frac{2z{D}_{p,q}{\mathcal{H}}_{\lambda }(z)}{{\mathcal{H}}_{\lambda }(z)-{\mathcal{H}}_{\lambda }(-z)}={\phi }_0\left(\frac{-(\lambda z+1)}{z(z+\lambda )}\right),{\mathcal{H}}_{\lambda }(0)={{\mathcal{H}}^{\prime }}_{\lambda }(0)-1=0.$$

It is a routine to verify that these functions belong to $\mathcal{S}\mathcal{T}{\mathcal{S}}_{p,q}{\phi }_0(z)$. When $\mu <{\sigma }_1$ or $\mu >{\sigma }_2$, equality holds if and only if f is ${\mathcal{F}}_2$ or one of its rotations. When ${\sigma }_1<\mu <{\sigma }_2$, equality holds if and only if f is ${\mathcal{F}}_3$ or one of its rotations. If $\mu ={\sigma }_1$, equality holds if and only if f is ${\mathcal{G}}_{\lambda }$ or one of its rotations and if $\mu ={\sigma }_2$, equality holds if and only if f is ${\mathcal{H}}_{\lambda }$ or one of its rotations.

Again replacing p by 1 and $q\to 1^{-}$ in Theorem 2.5, we get the following new result for the class $\mathcal{S}\mathcal{T}\mathcal{S}{\phi }_0(z)$:

Corollary 2.6.

Let $f\in \mathcal{S}\mathcal{T}\mathcal{S}{\phi }_0(z)$ and µ is any real number. If ${\sigma }_1=2(2-k)$, ${\sigma }_2=2(2+k)$ and ${\sigma }_3=4$, then

$$|a_3-\mu a_2^2|\le \left\{\begin{array}{cc}\frac{4-\mu }{4k^2},& \text{if}\mu \le {\sigma }_1,\\ \frac{1}{2k},& \text{if}{\sigma }_1\le \mu \le {\sigma }_2,\\ \frac{\mu -4}{4k^2},& \text{if}\mu \ge {\sigma }_2.\end{array}\right.$$

Further, if ${\sigma }_1\le \mu \le {\sigma }_3$, then

$$|a_3-\mu a_2^2|+(\mu -2(2-k))|a_2{|}^2\le \frac{1}{2k}.$$

If ${\sigma }_3\le \mu \le {\sigma }_2$, then

$$|a_3-\mu a_2^2|+(2(2+k)-\mu )|a_2{|}^2\le \frac{1}{2k}.$$

Sharpness holds for all the inequalities.

Theorem 2.7.

Let $0.57735\approx 1/\sqrt{3}<q\le p\le 1$, $f\in \mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and µ is any complex number. Then

$$|a_3-\mu a_2^2|\le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}\max \{1,|\frac{{[3]}_{p,q}({[3]}_{p,q}-1)\mu -2{[2]}_{p,q}^4}{k{[2]}_{p,q}^4}|\}.$$

The result is sharp.

Proof. Since $f\in \mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$, it follows that

(2.9)$$\frac{2D_{p,q}(z{D}_{p,q}f(z))}{D_{p,q}(f(z)-f(-z))}={\phi }_0(u(z)),$$

where $u:\mathbb{D}\to \mathbb{D}$ is a Schwarz function with u(0)=0. Define $\psi :\mathbb{D}\to ℂ$ by

$$\psi (z)=\frac{1+u(z)}{1-u(z)}=1+c_{1}z+c_2{z}^2+\mathrm{...}$$

or equivalently

$$u(z)=\frac{\psi (z)-1}{\psi (z)+1}=\frac{1}{2}(c_{1}z+(c_2-\frac{c_1^2}{2})z^2+\frac{1}{4}(c_1^3-4c_1{c}_2+4c_3)z^3+\mathrm{...}).$$

Then ψ is analytic in $\mathbb{D}$ with $\psi (0)=1$. Since $u:\mathbb{D}\to \mathbb{D}$, the function ψ has positive real part in $\mathbb{D}$, and hence $|c_i|\le 2$ for $i\ge 1$. Therefore we have

(2.10)$${\phi }_0(u(z))=1+\frac{1}{2k}{c}_{1}z+(\frac{1}{2k}(c_2-\frac{c_1^2}{2})+\frac{1}{2k^2}{c}_1^2)z^2+\cdots .$$

Using (2.9) and (2.10), we get

$${[2]}_{p,q}^2{a}_2=\frac{1}{2k}{c}_1$$ $${[3]}_{p,q}({[3]}_{p,q}-1)a_3=\frac{1}{2k}(c_2-\frac{c_1^2}{2})+\frac{1}{2k^2}{c}_1^2.$$

Since $1/\sqrt{3}<q\le p\le 1$, it follows that ${[3]}_{p,q}-1=p^2+pq+q^2-1\ge 3q^2-1>0$.

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

(2.11)$$\begin{array}{l}{a}_3-\mu a_2^2=\frac{1}{{[3]}_{p,q}({[3]}_{p,q}-1)}(\frac{1}{2k}(c_2-\frac{c_1^2}{2})+\frac{1}{2k^2}{c}_1^2)-\mu \frac{c_1^2}{4k^2{[2]}_{p,q}^4}\\ =\frac{1}{2k{[3]}_{p,q}({[3]}_{p,q}-1)}(c_2-\nu c_1^2),\end{array}$$

where

(2.12)$$\nu =\frac{{[3]}_{p,q}({[3]}_{p,q}-1)\mu -(2-k){[2]}_{p,q}^4}{2k{[2]}_{p,q}^4}.$$

Using Lemma 2.2, we get

$$|a_3-\mu a_2^2|\le 2\max \{1,|2\nu -1|\}.$$

Substituting the value of $|2\nu -1|$ completes the proof. The result is sharp for the functions

$$\frac{2D_{p,q}(z{D}_{p,q}f(z))}{D_{p,q}(f(z)-f(-z))}\prec {\phi }_0(z^2)\text{and}\frac{2D_{p,q}(z{D}_{p,q}f(z))}{D_{p,q}(f(z)-f(-z))}\prec {\phi }_0(z).$$

Replacing p by 1 and $q\to 1^{-}$ in Theorem 2.7, we get the following new result for the class $\mathcal{C}\mathcal{S}{\phi }_0(z)$.

Corollary 2.8.

If the function f is in the class $\mathcal{C}\mathcal{S}{\phi }_0(z)$ and $\mu$ a complex number, then

$$|a_3-\mu a_2^2|\le \frac{1}{6k}\max \left\{1,\left|\frac{3\mu -16}{8k}\right|\right\}.$$

The result is sharp.

Theorem 2.9.

Let $0.57735\approx 1/\sqrt{3}<q\le p\le 1$, $f\in \mathcal{C}{\mathcal{S}}_{p,q}{\phi }_0(z)$ and µ is any real number and let

$${\delta }_1=\frac{(2-k){[2]}_{p,q}^4}{{[3]}_{p,q}({[3]}_{p,q}-1)},{\delta }_2=\frac{(2+k){[2]}_{p,q}^4}{{[3]}_{p,q}({[3]}_{p,q}-1)}\text{and}{\delta }_3=\frac{2{[2]}_{p,q}^4}{{[3]}_{p,q}({[3]}_{p,q}-1)}.$$

Then

(2.13)$$|a_3-\mu a_2^2|\le \left\{\begin{array}{cc}\frac{2{[2]}_{p,q}^4-\mu {[3]}_{p,q}({[3]}_{p,q}-1)}{k^2{[3]}_{p,q}({[3]}_{p,q}-1){[2]}_{p,q}^4},& \mu \le {\delta }_{\text{1}}\\ \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)},& {\delta }_{\text{1}}\le \mu \le {\delta }_{\text{2}}\\ \frac{{[3]}_{p,q}({[3]}_{p,q}-1)\mu -2{[2]}_{p,q}^4}{k^2{[3]}_{p,q}({[3]}_{p,q}-1){[2]}_{p,q}^4},& \mu \ge {\delta }_{\text{2}}\end{array}\right.$$

Moreover, if ${\delta }_1\le \mu \le {\delta }_3$, then

(2.14)$$|a_3-\mu a_2^2|+\frac{{[3]}_{p,q}({[3]}_{p,q}-1)\mu -(2-k){[2]}_{p,q}^4}{{[3]}_{p,q}({[3]}_{p,q}-1)}|a_2{|}^2\le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)},$$

and if ${\delta }_3\le \mu \le {\delta }_2$, then

(2.15)$$|a_3-\mu a_2^2|+\frac{(2+k){[2]}_{p,q}^4-{[3]}_{p,q}({[3]}_{p,q}-1)\mu }{{[3]}_{p,q}({[3]}_{p,q}-1)}|a_2{|}^2\le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}.$$

These results are sharp.

Proof. Using Lemma 2.3, (2.11), and the fact ${[3]}_{p,q}-1=p^2+pq+q^2-1\ge 3q^2-1>0$ for $1/\sqrt{3}<q\le p\le 1$, (2.12), we get the following results.

If $\mu \le {\delta }_1$, then

$$|a_3-\mu a_2^2|\le \frac{2{[2]}_{p,q}^4-\mu {[3]}_{p,q}({[3]}_{p,q}-1)}{k^2{[3]}_{p,q}({[3]}_{p,q}-1){[2]}_{p,q}^4}.$$

If ${\delta }_1\le \mu \le {\delta }_2$, then

$$|a_3-\mu a_2^2|\le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}.$$

If $\mu \ge {\delta }_2$, then

$$|a_3-\mu a_2^2|\le \frac{{[3]}_{p,q}({[3]}_{p,q}-1)\mu -2{[2]}_{p,q}^4}{k^2{[3]}_{p,q}({[3]}_{p,q}-1){[2]}_{p,q}^4}.$$

Since $1/\sqrt{3}<q\le p\le 1$, it follows that ${[3]}_{p,q}-1=p^2+pq+q^2-1\ge 3q^2-1>0$. On the other hand, using (2.11) and (2.12) for the values ${\delta }_1\le \mu \le {\delta }_3$ we have

$$\begin{array}{l}|a_3-\mu a_2^2|+(\mu -{\delta }_1)|a_2{|}^2=\frac{1}{2k{[3]}_{p,q}({[3]}_{p,q}-1)}|c_2-\nu c_1^2|\\ +(\mu -\frac{(2-k){[2]}_{p,q}^4}{{[3]}_{p,q}({[3]}_{p,q}-1)})\frac{c_1^2}{4k^2{[2]}_{p,q}^4}\\ =\frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}(\frac{1}{2}|c_2-\nu c_1^2|+\nu c_1^2)\\ \le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}.\end{array}$$

Similarly, if ${\delta }_3\le \mu \le {\delta }_2$ we have

$$|a_3-\mu a_2^2|+({\delta }_2-\mu )|a_2{|}^2\le \frac{1}{k{[3]}_{p,q}({[3]}_{p,q}-1)}.$$

Similar to the results in previous corollaries, putting p=1 and $q\to 1^{-}$, the Fekete-Szegö functionals can be obtained for the class $\mathcal{C}\mathcal{S}{\phi }_0(z)$.