Applications of fractional derivatives for Alexander integral operator
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Applications of fractional derivatives for Alexander integral operator Hatun Özlem Güney1
· Mugur Acu2 · Daniel Breaz3 · Shigeyoshi Owa4
Received: 12 July 2020 / Accepted: 7 October 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Let Tn be the class of functions f (z) = z + an+1 z n+1 + an+2 z n+2 + . . . that are analytic in the closed unit disc U. With m different boundary points z s , (s = 1, 2, . . . , m), we consider αm ∈ eiβ A j+λ f (U), here A j+λ is given by using fractional derivatives D j+λ f (z) for f (z) ∈ Tn . Using A j+λ , we introduce a subclass Pn (αm , β, ρ; j, λ) of Tn . The main goal of our paper is to discuss some interesting results of f (z) in the class Pn (αm , β, ρ; j, λ). Keywords Analytic function · Alexander integral operator · Fractional derivative · Fractional integral · Gamma function · Miller and Mocanu Lemma Mathematics Subject Classification Primary 30C45
1 Introduction Let Tn be the class of functions f (z) = z +
∞
ak z k , n ∈ N = {1, 2, 3, . . .}
(1.1)
k=n+1
B
Hatun Özlem Güney [email protected] Mugur Acu [email protected]; [email protected] Daniel Breaz [email protected] Shigeyoshi Owa [email protected]
1
Department of Mathematics, Faculty of Science, University of Dicle, Diyarbakır, Turkey
2
Department of Mathematics, University “Lucian Blaga” of Sibiu, Str. Dr. I. Ratiu, No. 5-7, 550012 Sibiu, Romania
3
Department of Exact Sciences and Engineering, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania
4
Honorary Professor “1 Decembrie 1918”, University Alba Iulia, Alba Iulia, Romania
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H. Ö. Güney et al.
that are analytic in the closed unit disc U = {z ∈ C : |z| ≤ 1}. For f (z) ∈ Tn , Alexander [2] had defined the following Alexander integral operator A−1 f (z) given by z ∞ f (t) ak k (1.2) A−1 f (z) = dt = z + z . t k 0 k=n+1
The above Alexander integral operator was applied for some subclasses of analytic functions in the open unit disc U = {z ∈ C : |z| < 1} by Acu [1], by Kugita et al. [5] and by Güney and Owa [3]. Recently, Güney and Owa [3] consider A− j f (z) = A− j+1 (A−1 f (z)) = z +
∞ ak k z , kj
j ∈N
(1.3)
k=n+1
where A0 f (z) = f (z). From the among various definitions of f (z) ∈ Tn for fractional calculus (that is, fractional derivatives and fractional integrals) given in the literature, we would like to recall here the following definitions for fractional calculus which were used by Owa [8] and Owa and Srivastava [9]. Definition 1.1 The fractional integral of order λ for f (z) ∈ Tn is defined by z f (t) 1 dt , (λ ≥ 0) Dz−λ f (z) = (λ) 0 (z − t)1−λ
(1.4)
where the multiplicity of (z − t)λ−1 is removed by requiring log(z − t) to be real when z − t > 0 and is the Gamma function. With the above definition, we know that ∞ k! 1 z 1+λ + ak z k+λ (2 + λ) (k + 1 + λ)
Dz−λ f (z) =
(1.5)
k=n+1
for λ ≥ 0 and f (z) ∈ Tn . Definition 1.2 The fractional derivative of order λ for f (z) ∈ Tn is defined by z d f (t) d 1 dt, (0 ≤ λ < 1)
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