Subordination results for analytic functions associated with fractional q -calculus operators with complex order
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Subordination results for analytic functions associated with fractional q-calculus operators with complex order M. K. Aouf1 · A. O. Mostafa1 Received: 24 March 2020 / Accepted: 25 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Using a q-fractional calculus operators, we define the classes S α (λ, β, b, q) and G α (λ, β, b, q) of analytic functions with complex order and investigate some subordination results for these classes. Keywords q-Fractional calculus operators · Analytic functions · Complex order · Factor sequence Mathematics Subject Classification 30C45
1 Introduction The class of analytic functions of the form: f (z) = z +
∞
ak z k (z ∈ U = {z ∈ C : |z| < 1}),
(1.1)
k=n+1
is denoted by A(n) and A(1) = A. Also, denote by K the subclass of A consisting of functions g which are convex in U. In this investigation, by making use of various operators of q−calculus and fractional q-calculus, we recall the following definitions and notations. The q-shifted factorial is defined for α, q ∈ C (|q| < 1) and n ∈ N0 = N ∪ {0}, N = {1, 2, . . .} by: 1, n=0 (α; q)n = (1.2) (1 − α)(1 − αq) . . . (1 − αq n−1 ), n ∈ N
B
M. K. Aouf [email protected] A. O. Mostafa [email protected]
1
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
123
M. K. Aouf, A. O. Mostafa
and in terms of the basic (or q-)gamma function q (z), we have q (α + n)(1 − q)n (n ∈ N ), q (α)
(1.3)
(q, q)∞ (1 − q)1−α (0 < q < 1). (q α ; q)∞
(1.4)
(q α ; q)n = where (see for example [14, p. 6]) q (α) = We note that
(α; q)∞ =
∞
(1 − αq k ) (|q| < 1).
k=0
It is known that (see [14]) q (1 + α) = [α]q q (α),
(1.5)
where [α]q denotes the basic (or q-)number defined by [α]q =
1 − qα (0 < q < 1), 1−q
(1.6)
which readily yields [α]q =
1 − qα → α as q → 1− . 1−q
It is also known, for the classical (Euler’s) gamma function (z), that q (z) → (z) as q → 1− . In view of the relation lim
q→1−
(q α ; q)k = (α)k , (1 − q)k
we observe that the q-shifted factorial (1.2) reduces to the familiar Pochhammer symbol (α)k , (α)k = α(α + 1) . . . (α + k − 1). For a fixed μ ∈ C, a set D is called a μ-geometric set, if for z ∈ D, μz ∈ D. For f defined on a q−geometric set, Jackson’s q-derivative and q-integral (0 < q < 1) of a function on a subset of C are, respectively, given by (see [2,5,8,14–16,23,24,30,33]) f (z)− f (qz) (1−q)z , z = 0 Dq f (z) = (1.7) f (0) z=0 and z f (t)dq t = z(1 − q)
∞
q k f (zq k ).
k=0
0
In particular, when f (z) = z k (k ∈ N ) in (1.7), the q− derivative is given by Dq z k =
123
z k − (qz)k = [k]q z k−1 . (1 − q)z
(1.8)
Subordination results for analytic functions associated…
Now, for a function f (z), we redefine the fractional q-calculus operators, which were recently studied by Purohit and Raina [20,21] (see also [1]). λ of order Definition 1 (Fractional q-integral operator) The fractional q−integral operator Iq,z λ is defined for a function f (z) by λ Iq,z
f (z) =
−λ Dq,z
1
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