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:

which are analytic in the open punctured unit disc

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

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

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(0≤β<1). Let denote the class of functions f(z) of the form

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

and the class consists of functions for which

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

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

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])

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

where

As q → 1⁻, [k]_q → k, we have

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

We note that:

• (i) limq→1- ℱq,α*(ϕ)=ℱα*(ϕ) (see Aouf et al. [4]);

• (ii) limq→1- ℱq,0*(ϕ)=Σ*(ϕ) (see Silverman et al. [18] and Ali and Ravichandran [2]);

• (iii) limq→1- ℱq,0*(1+z1-z)=ℱ*(1)=ℱ* (see Aouf [3, with b = 1]);

• (iv) limq→1- ℱq,0*(1+(1-2β)z1-z)=Σ*(β) (0≤β<1) (see Pommerenke [16]);

• (v) limq→1- ℱq,0*(1+β(1-2γη)z1+β(1-2γ)z)=Σ(η,β,γ) (0≤η<1, 0<β≤1,12≤γ≤1) (see Kulkarni and Joshi [12]);

• (vi) limq→1- ℱq,0*(1+Az1+Bz)=K1(A,B)         (0≤B<1, -B<A<B) (see Karunakaran [11]).

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

The result is sharp for the functions given by

Lemma 2.([13])

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

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

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

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

and

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 ℱq,α*(ϕ) and μ is a complex number, then

and

The result is sharp.

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 ℱq,α*(ϕ), then

where

The result is sharp. Further, let

(i) If σ₁ ≤ μ ≤ σ₃, then

(ii) If σ₃ ≤ μ ≤ σ₂, then

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].

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

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

In terms of the analogue of the gamma function

where the q–gamma function is defined by

We note that

where

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

Also, let us define

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

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

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 ℱq,α,v*(ϕ) and μ is a complex number, then

• (i) ∣a1-μa02∣≤42(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|}         (B1≠0),

• (ii)    ∣a1∣≤42(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 + ..., (B_i > 0, i ∈ {1, 2}, α > 0). If f(z) given by (1.1) belongs to the class ℱq,α,v*(ϕ), then

where

and

The result is sharp. Further, let

(i) If σ1*≤μ≤σ3*, then

(ii) If σ3*≤μ≤σ2*, then