Characterizations of some fractional-order operators in complex domains and their extensive implications to certain analytic functions

Abstract

The aim of this research note is firstly to introduce certain fundamental information in relation with various operators of fractional-order calculus in the complex plane, then create some comprehensive results associating with certain analytic functions as implications of those operators, and also present numerous conclusions and recommendations for the related researchers.

Various fractional-order type operators and some of their implications to certain normalized functions analytic in the open unit disc

“…can be easily established (as in the similar determinations constituted in the references given by [4], [10] and [11]).…”

“…where α ∈ R * and w 0 ∈ U. Then, by making use of the assertion presented by (10) of Lemma 1 and also by considering the following relationships:…”

“…is also encountered as the Srivastava-Owa fractional derivative operator of order s + ω (0 ≤ ω < 1; s ∈ N 0 ) in the mathematical literature. For more information associating with the details of both differ-integral operators define by (3) together with (4), one may refer to the main works in [13], [16] and [18], and, for various examples, it can be also checked the results in the (recent) studies presented by [1], [7] and [10].…”

On Classes of Non-Carathéodory Functions Associated with a Family of Functions Starlike in the Direction of the Real Axis

“…Motivated by [11][12][13][14][15][16][17], we now define a generalized class of Bazilevič functions.…”

“…< 1, which implies the function Λ[α, β; ψ(z)] ∈ P (see Theorem 2 in [27]). Now, f ∈ MS ν (α, β; γ; ψ(z)) (z ∈ U ) implies that there is a Schwarz function w(z) such that (16) Define the function h(z) by…”