Kyungpook Mathematical Journal 2018; 58(2): 307-318  
On the Fekete-Szegö Problem for a Certain Class of Meromorphic Functions Using q-Derivative Operator
Mohamed Kamal Aouf and Halit Orhan*
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, e-mail: mkaouf127@yahoo.com, Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240, Turkey, e-mail: orhanhalit607@gmail.com
*Corresponding Author.
Received: October 6, 2017; Accepted: March 29, 2018; Published online: June 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In this paper, we obtain Fekete-Szegö inequalities for certain class of meromorphic functions f(z) for which -(1-αq)qzDqf(z)+αqzDq[zDqf(z)](1-αq)f(z)+αzDqf(z)ϕ(z)(α(0,1],0<q<1).

Sharp bounds for the Fekete-Szegö functional a1-μa02 are obtained.

Keywords: Analytic, meromorphic, q-starlike and convex functions, Fekete-Szegö problem, convolution
1. Introduction

The theory of q–analysis has important role in many areas of mathematics and physics, for example, in the areas of ordinary fractional calculus, optimal control problems, q–difference, q–integral equations and in q–transform analysis (see for instance [1, 6, 8, 9]). The study of q–calculus has gained momentum years mainly due to the pioneer work of M. E. H. Ismail et al. [7] in recent years; it was followed by such works as those by S. Kanas and D. Raducanu [10] and S. Sivasubramanian and M. Govindaraj [19]. Let Σ denote the class of meromorphic functions of the form:

f(z)=1z+k=0akzk,

which are analytic in the open punctured unit disc

U*={z:zand 0<z<1}=U{0}.

A function f ∈ Σ is meromorphic starlike of order β, denoted by Σ*(β), if

-{zf(z)f(z)}>β(0β<1;zU).

The class Σ*(β) was introduced and studied by Pommerenke [16] (see also Miller [14]). Let ϕ(z) be an analytic function with positive real part on satisfies ϕ(0) = 1 and ϕ′(0) > 0 which maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. Let Σ*(ϕ) be the class of functions f(z) ∈ Σ for which

-zf(z)f(z)ϕ(z)(zU).

The class Σ*(ϕ) was introduced and studied by Silverman et al. [18]. The class Σ*(β) is the special case of Σ*(ϕ) when ϕ(z)=1+(1-2β)z1-z(0β<1). Let denote the class of functions f(z) of the form

f(z)=z+k=2akzk,

which are analytic in the open unit disc and let be the subclass of consisting of functions which are analytic and univalent in . Ma and Minda [13] introduced and studied the class which consists of functions for which

zf(z)f(z)ϕ(z)(zU),

and the class consists of functions for which

1+zf(z)f(z)ϕ(z)(zU).

Following Ma and Minda [13], Shanmugam and Sivasubramanian [17] defined a more general class ℳα(ϕ) consists of functions for which

zf(z)+αz2f(z)(1-α)f(z)+αzf(z)ϕ(z)(α0).

Analogous to the class ℳα(ϕ), Aouf et al. [4] defined the class α*(ϕ) as follows: For α ∈ ℂ(0, 1], let α*(ϕ) be the subclass of Σ consisting of functions f(z) of the form (1.1) and satisfying the analytic criterion:

-zf(z)+αz2f(z)(1-α)f(z)+αzf(z)ϕ(z).

For a function f(z) ∈ Σ given by (1.1) and 0 < q < 1, the q–derivative of a function f(z) is defined by (see Gasper and Rahman [6])

Dqf(z)=f(qz)-f(z)(q-1)zif zU*.

From (1.2), we deduce that Dqf(z) for a function f(z) of the form (1.1) is given by

Dqf(z)=-1qz2+k=0[k]qakzk-1(z0),

where

[i]q=1-qi1-q.

As q → 1, [k]qk, we have

limq1-Dqf(z)=f(z).

Making use of the q–derivative Dq, we introduce the subclass q,α*(ϕ) as follows: For α ∈ ℂ(0, 1], 0 < q < 1, a function f(z) ∈ Σ is said to be in the class q,α*(ϕ), if and only if

-(1-αq)qzDqf(z)+αqzDq[zDqf(z)](1-αq)f(z)+αzDqf(z)ϕ(z)(zU).

We note that:

(i) limq1-q,α*(ϕ)=α*(ϕ) (see Aouf et al. [4]);

(ii) limq1-q,0*(ϕ)=Σ*(ϕ) (see Silverman et al. [18] and Ali and Ravichandran [2]);

(iii) limq1-q,0*(1+z1-z)=*(1)=* (see Aouf [3, with b = 1]);

(iv) limq1-q,0*(1+(1-2β)z1-z)=Σ*(β)(0β<1) (see Pommerenke [16]);

(v) limq1-q,0*(1+β(1-2γη)z1+β(1-2γ)z)=Σ(η,β,γ)(0η<1,0<β1,12γ1) (see Kulkarni and Joshi [12]);

(vi) limq1-q,0*(1+Az1+Bz)=K1(A,B)         (0B<1,-B<A<B) (see Karunakaran [11]).

2. Fekete-Szegö Problem

To prove our results, we need the following lemmas.

Lemma 1.([13])

If p(z) = 1+c1z + c2z2 + ⋯ is a function with positive real part inand μ is a complex number, then

c2-μc122max{1;2μ-1}.

The result is sharp for the functions given by

p(z)=1+z21-z2andp(z)=1+z1-z.

Lemma 2.([13])

If p1(z) = 1+c1z + c2z2 + … is a function with positive real part in, then

c2-νc12{-4ν+2ifν0,2if0ν1,4ν-2ifν1.

When ν < 0 or ν > 1, the equality holds if and only if p1(z)=1+z1-z or one of its rotations. If 0 < ν < 1, then the equality holds if and only if p1(z)=1+z21-z2 or one of its rotations. If ν = 0, the equality holds if and only if

p1(z)=(12+λ2)1+z1-z+(12-λ2)1-z1+z(0λ1),

or one of its rotations. If ν = 1, the equality holds if and only if

1p1(z)=(12+λ2)1+z1-z+(12-λ2)1-z1+z(0λ1),

or one of its rotations. Also the above upper bound is sharp and it can be improved as follows when 0 < ν < 1:

c2-νc12+νc122(0<ν12),

and

c2-νc12+(1-ν)c122(12<ν<1).

Unless otherwise mentioned, we assume throughout this paper that α ∈ ℂ(0, 1] and 0 < q < 1.

Theorem 1

Let ϕ(z) = 1+B1z +B2z2 +…. If f(z) given by (1.1) belongs to the class q,α*(ϕ) and μ is a complex number, then

(i)         a1-μa0211+q|(q-2α)B1(q-α+αq)|×max {1,|B2B1-[1-μ(q-2α)(q-α+αq)(q+1)(q-α)2]B1|}         (B10),

and

(ii)         a111+q|(q-2α)B2(q-α+αq)|         (B1=0).

The result is sharp.

Proof

If f(z)α*(ϕ), then there is a Schwarz function w(z) in with w(0) = 0 and |w(z)| < 1 in and such that

-(1-αq)qzDqf(z)+αqzDq[zDqf(z)](1-αq)f(z)+αzDaf(z)=ϕ(w(z)).

Define the function p1(z) by

p1(z)=1+w(z)1-w(z)=1+c1z+c2z2+.

Since w(z) is a Schwarz function, we see that ℜ{p1(z)} > 0 and p1(0) = 1. Define

p(z)=-(1-αq)qzDqf(z)+αqzDq[zDqf(z)](1-αq)f(z)+αzDqf(z)=1+b1z+b2z2+.

In view of (2.3), (2.4) and (2.5), we have

p(z)=ϕ(p1(z)-1p1(z)+1).

Since

p1(z)-1p1(z)+1=12[c1z+(c2-c122)z2+(c3+c134-c1c2)z3+].

Therefore, we have

ϕ(p1(z)-1p1(z)+1)=1+12B1c1z+[12B1(c2-c122)+14B2c12]z2+,

and from this equation and (2.6), we obtain

b1=12B1c1,

and

b2=12B1(c2-c122)+14B2c12.

Then, from (2.5) and (1.1), we see that

b1=-(q-αq-2α)a0,

and

b2=(q-αq-2α)2a02-(q+1)(q-α+αq)q-2αa1,

or, equivalently, we have

a0=-(q-2αq-α),

and

a1=-(q-2α)B12(1+q)(q-α+αq)[c2-c122(1-B2B1+B1)].

Therefore

a1-μa02=-(q-2α)B12(1+q)(q-α+αq){c2-νc12},

where

ν=12[1-B2B1+B1-μ(q-2α)(q-α+αq)(q+1)B1(q-α)2].

Now, the result (2.1) follows by an application of Lemma 1. Also, if B1 = 0, then

a0=0and a1=-(q-2α)B2c124(1+q)(q-α+αq).

Since p(z) has positive real part, |c1| ≤ 2 (see Nehari [15]), so that

a111+q|(q-2α)B2(q-α+αq)|,

this proving (2.2). The result is sharp for the functions

-(1-αq)qzDqf(z)+αqzDq[Dqf(z)](1-αq)f(z)+αzDqf(z)=ϕ(z2),

and

-(1-αq)qzDqf(z)+αqzDq[Dqf(z)](1-αq)f(z)+αzDqf(z)=ϕ(z).

This completes the proof of Theorem 1.

Remark 1

(i) For q → 1 in Theorem 1, we obtain the result obtained by Aouf et al. [4, Theorem 2.1];

(ii) For q → 1 and α = 0 in Theorem 1, we obtain the result obtained by Silverman et al. [18, Theorem 2.1].

By using Lemma 2, we can obtain the following theorem.

Theorem 2

Let ϕ(z)=1+B1z+B2z2+(Bi>0,i{1,2},0<α<q1+q).

If f(z) given by (1.1) belongs to the class q,α*(ϕ), then

a1-μa02{(q-2α)B12(1+q)(q-α+αq){-B2+[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B12}ifμσ1,(q-2α)B1(1+q)(q-α+αq)ifσ1μσ2,(q-2α)B12(1+q)(q-α+αq){B2-[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B12}ifμσ2,

where

σ1=(q-α)2[-B1-B2+B12](q-2α)(q-α+αq)(1+q)B12andσ2=(q-α)2(B1-B2+B12)(q-2α)(q-α+αq)(1+q)B12.

The result is sharp. Further, let

σ3=(q-α)2[-B2+B12](q-2α)(q-α+αq)(1+q)B12.

(i) If σ1μσ3, then

a1-μa02+(q-α)2(q-2α)(q-α+αq)(1+q)B12×{(B1+B2)+[μ(1+q)(q-2α)(q-α+αq)q(1-α)2-1]B12}a02(q-2α)B1(1+q)(q-α+αq).

(ii) If σ3μσ2, then

a1-μa02+(q-α)2(q-2α)(q-α+αq)(1+q)B12×{(B1-B2)+[1-μ(1+q)(q-2α)(q-α+αq)q(1-α)2]B12}a02(q-2α)B1(1+q)(q-α+αq).
Proof

First, let μσ1. Then

a1-μa02(q-2α)B1(1+q)(q-α+αq){-B2B1+[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B1}(q-2α)B12(1+q)(q-α+αq){-B2+[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B12}.

Let, now σ1μσ2. Then, using the above calculations, we obtain

a1-μa02(q-2α)B1(1+q)(q-α+αq).

Finally, if μσ2, then

a1-μa02(q-2α)B1(1+q)(q-α+αq){B2B1-[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B1}(q-2α)B12(1+q)(q-α+αq){B2-[1-μ[q-α(1+q)](1+q)(q-2α)q(1-α)2]B12}.

To show that the bounds are sharp, we define the functions Kϕn (n ≥ 2) by

-(1-αq)qzDqKϕn(z)+αqzDq[zDqKϕn(z)](1-αq)Kϕn(z)+αzDqKϕn(z)=ϕ(zn-1),z2Kϕn(z)z=0=0=-z2Kϕn(z)z=0-1,

and the functions Fγ and Gγ (0 ≤ γ ≤ 1) by

-(1-αq)qzDqFγ(z)+αqzDq[zDqFγ(z)](1-αq)Fγ(z)+αzDqFγ(z)=ϕ(z(z+γ)1+γz),z2Fγ(z)z=0=0=-z2Fγ(z)z=0-1,

and

-(1-αq)qzDqGγ(z)+αqzDq[zDqGγ(z)](1-αq)Gγ(z)+αzDqGγ(z)=ϕ(-z(z+γ)1+γz),z2Gγ(z)z=0=0=-z2Gγ(z)z=0-1.

Cleary the functions Kϕn, Fγ and Gγq,α*(ϕ). Also we write Kϕ = Kϕ2. If μ < σ1 or μ > σ2, then the equality holds if and only if f(z) is Kϕ or one of its rotations. When σ1< μ < σ2, then the equality holds if f(z) is Kϕ3 or one of its rotations. If μ = σ1, then the equality holds if and only if f(z) is Fγ or one of its rotations. If μ = σ2, then the equality holds if and only if f(z) is Gγ or one of its rotations. This completes the proof of Theorem 2.

Remark 2

(i) For q → 1 in Theorem 2, we obtain the result obtained by Aouf et al. [4, Theorem 2];

(ii) Putting q → 1 and α = 0 in Theorem 2, we obtain the result obtained by Ali and Ravichandran [2, Theorem 5.1].

3. Applications to Functions Defined by q–Bessel Function

We recall some definitions of q–calculus which we will be used in our paper. For any complex number α, the q–shifted factorials are defined by

(α;q)0=1;(α;q)n=k=0n-1(1-αqk)(n={1,2,}).

If |q| < 1, the definition (3.1) remains meaningful for n = ∞ as a convergent infinite product

(α;q)=j=0(1-αqj).

In terms of the analogue of the gamma function

(qα;q)n=Γq(α+n)(1-q)nΓq(α)(n>0),

where the q–gamma function is defined by

Γq(x)=(q;q)(1-q)1-x(qx;q)(0<q<1).

We note that

limq1-(qα;q)n(1-q)n=(α)n,

where

(α)n={1if n=0,α(α+1)(α+2)(α+n-1)if n.

Now, consider the q–analoge of Bessel fnction defined by (Jackson [8])

Jv(1)(z;q)=(qv+1;q)(q;q)k=0(-1)k(q;q)k(qv+1;q)k(z2)2k+ν(0<q<1).

Also, let us define

v(z;q)=2v(q;q)(qv+1;q)(1-q)vzv/2+1Jv(1)(z1/2(1-q);q)=1z+k=0(-1)k+1(1-q)2(k+1)4(k+1)(q;q)k+1(qv+1;q)k+1zk(zU).

By using the Hadamard product (or convolution), we define the linear operator ℒq,υ : ∑ → ∑, as follows:

(q,vf)(z)=v(z;q)*f(z)=1z+k=0(-1)k+1(1-q)2(k+1)4(k+1)(q;q)k+1(qv+1;q)k+1akzk.

As q → 1, the linear operator ℒq,υ reduces to the operator ℒυ introduced and studied by Aouf et al. [5]. For 0 < q < 1 and α ∈ ℂ(0, 1], let q,α,v*(ϕ) be the subclass of ∑ consisting of functions f(z) of the form (1.1) and satisfies the analytic criterion:

-(1-αq)qzDq(q,vf)+αqzDq[Dq(q,vf)](1-αq)(q,vf)+αzDq(q,vf)ϕ(z)(zU).

Using similar arguments to those in the proof of the above theorems, we obtain the following theorems.

Theorem 3

Let ϕ(z) = 1+B1z + B2z2 + · · ·. If f(z) given by (1.1) belongs to the class q,α,v*(ϕ) and μ is a complex number, then

(i) a1-μa0242(1-qv+1)(1-qv+2)(1-q)2|B1(q-2α)q+αq-α|×max {1,|B2B1-[1-μ(q-2α)(1-qv+1)(q-α+αq)(1-qv+2)(q-α)2]B1|}         (B10),

(ii)    a142(1-qv+1)(1-qv+2)(1-q)2|B2(q-2α)q+αq-α|         (B1=0).

The result is sharp.

Theorem 4

Let ϕ(z) = 1+B1z + B2z2 + ..., (Bi > 0, i ∈ {1, 2}, α > 0). If f(z) given by (1.1) belongs to the class q,α,v*(ϕ), then

a1-μa02{42(1-qv+1)(1-qv+2)(q-2α)B12(1-q)2(q+αq-α)×{-B2+[1-μ(q-2α)(1-qv+1)(q-α+αq)(1-qv+2)(q-α)2]B12}ifμσ1*,42(1-qv+1)(1-qv+2)[q-α(q+1)]B1q(1-q)2ifσ1*μσ2*,42(1-qv+1)(1-qv+2)[q-α(q+1)]q(1-q)2×{B2-[1-μ(q-2α)(1-qv+1)(q-α+αq)(1-qv+2)(q-α)2]B12}ifμσ2*,

where

σ1*=(q-α)2(1-qv+2)[-B1-B2+B12](q-2α)(q-α+αq)(1-qv+1)B12,

and

σ2*=(q-α)2(1-qv+2)[B1-B2+B12](q-2α)(q-α+αq)(1-qv+1)B12.

The result is sharp. Further, let

σ3*=(q-α)2(1-qv+2)[-B2+B12](q-2α)(q-α+αq)(1-qv+1)B12.

(i) If σ1*μσ3*, then

a1-μa02+(1-qv+2)(q-α)2(q-2α)(q-α+αq)(1-qv+1)B12×{(B1+B2)+[μ(q-2α)(1-qv+1)(1-qv+2)(q-α+αq)(1-qv+2)(q-α)2-1]B12}a0242(q-2α)(1-qv+1)(1-qv+2)B1(1-q)2(q-α+αq).

(ii) If σ3*μσ2*, then

a1-μa02+(1-qv+2)(q-α)2(q-2α)(q-α+αq)(1-qv+1)B12×{(B1-B2)+[1-μ(q-2α)(1-qv+1)(1-qv+2)(q-α+αq)(1-qv+2)(q-α)2-1]B12}a0242(q-2α)(1-qv+1)(1-qv+2)B1(1-q)2(q-α+αq).
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