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),
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