Certain subclasses of analytic functions with varying arguments associated with q -difference operator

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Certain subclasses of analytic functions with varying arguments associated with q-difference operator M. K. Aouf1 · A. O. Mostafa1 · R. E. Elmorsy1 Received: 24 June 2020 / Accepted: 29 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020

Abstract In this paper, we introduce the new classes V E q (n, λ, β) and VG q (n, λ, β) of analytic functions with varying arguments defined by q-Al-Oboudi difference operator and study different properties for them. Keywords Analytic functions · Coefficient estimates · Distortion · q- Al-Oboudi difference operator · Varying arguments Mathematics Subject Classification 30C45

1 Introduction Denote by A be the class of analytic functions of the form f (z) = z +

∞ 

ak z k , z ∈ U = {z : z ∈ C : |z| < 1} .

(1.1)

k=2

For f (z) ∈ A, given by (1.1) and 0 < q < 1, the Jackson’s q−derivative of a function f is given by [18] (see also [2,3,8,12,14–16,19,22,23,25] and [27]): Dq f (z) =

f (z) − f (qz) , (0 < q < 1, z  = 0) , (1 − q) z

(1.2)



Dq f (0) = f (0) and Dq2 f (z) = Dq (Dq f (z)). From ( 1.2) we have Dq f (z) = 1 +

∞ 

[k]q ak z k−1 ,

(1.3)

k=2

B

M. K. Aouf [email protected] A. O. Mostafa [email protected] R. E. Elmorsy [email protected]

1

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

123

M. K. Aouf et al.

where [k]q =

1 − qk 1−q

(0 < q < 1).

(1.4)



As q → 1− , [k]q → k and, so Dq f (z) = f (z). For f (z) ∈ A, λ ≥ 0 and 0 < q < 1. Let 0 Dλ,q f (z) = f (z), 1 Dλ,q f (z) = (1 − λ) f (z) + λz Dq f (z) = Dλ,q f (z)

=z+

∞    1 + λ([k]q − 1) ak z k k=2

.. . n−1 n f (z) = Dλ,q (Dλ,q f (z)) Dλ,q

(n ∈ N, N = {1, 2, . . .}).

(1.5)

It follows from (1.5) and (1.1) that n Dλ,q f (z) = z +

∞  

n 1 + λ([k]q − 1) ak z k (n ∈ N0 = N ∪ {0}).

(1.6)

k=2

We note that n f (z) = D n f (z) (see Al-Oboudi [1] , Aouf and Mostafa [9] and Aouf et (i) limq−→1− Dλ,q λ al. [13] ); n f (z) = D n f (z) (see Govindaraj and Sivasubramanian [17] and Aouf et al. [11]); (ii) D1,q q n f (z) = D n f (z) see Salagean [21] (also see [4–6] and [7]). (iii) limq−→1− D1,q

Let E q (n, λ, β) and G q (n, λ, β) denote the subclasses of A consisting of functions f (z) which satisfy the inequalities   n f (z)) z Dq (Dλ,q ReRe < β, (1.7) n f (z) Dλ,q or, equivalently,

  n f (z)) z Dq (Dλ,q   − 1   n Dλ,q f (z)    < 1 (β > 1),  z D (D n f (z))   q λ,q  D n f (z) − (2β − 1) 

(1.8)

λ,q



and ReRe or equivalently,

      

n f (z))) Dq (z Dq (Dλ,q n f (z)) Dq (Dλ,q

< β,

     < 1 (β > 1).  − (2β − 1) 

n f (z))) Dq (z Dq (Dλ,q n f (z)) Dq (Dλ,q n f (z))) Dq (z Dq (Dλ,q n f (z)) Dq (Dλ,q



−1

It follows from (1.7) and (1.9) that n n Dλ,q f (z) ∈ G q (n, λ, β) ⇔ z Dq (Dλ,q f (z)) ∈ E q (n, λ, β).

We note that

123

(1.9)

(1.10)

Certain subclasses of analytic functions with...

(i) limq→1− E q (n, λ, β) = E(n, λ, β) :  ReRe or, equivalently,



z(Dλn f (z)) Dλn f (z)

 < β,

     z(Dλn f (z))   − 1 n Dλ f (z)    < 1 (β > 1),