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

(1.1)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 RE1+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

(1.2)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<q≤p≤1. 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<1, 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 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

Dqf(z)=f(z)−f(qz)(1−q)z,if 0<q<1,z≠0,1,if q→1−,z=0,f′(z),if q→1−,z≠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:

(1.3)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

(1.4)2(k2−keiθ)zDp,qf(z)−(k2+e2iθ)(f(z)−f(−z))≠0

In view of convolution, we obtain

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

and

(1.6)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

(1.7)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:

(1.8)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

(1.9)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:

(1.10)logf(z)z=2∑ n=1∞γnzn, z∈D.

The n^th 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 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 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<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 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<q≤p≤1 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

(2.1)|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

(2.2)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

(2.3)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

(2.4)[2]p,qa2=c112k, (2.5)([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 (2.6)+2k(−3+2k−2(k−2)[3]p,q)c1c2+4k2([3]p,q−1)c3.

Since 1/3<q≤p≤1, 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

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

where

(2.8)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<q≤p≤1, 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<q≤p≤1, 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<q≤p≤1, 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

(2.9)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

(2.10)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<q≤p≤1, 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

(2.11)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

(2.12)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

2Dp,q(zDp,qf(z))Dp,q(f(z)−f(−z))≺φ0(z2) and 2Dp,q(zDp,qf(z))Dp,q(f(z)−f(−z))≺φ0(z).  

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<q≤p≤1, 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

(2.13)|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

(2.14)|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

(2.15)|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<q≤p≤1, (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<q≤p≤1, 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).