tailieunhanh - Some subclasses of analytic functions of complex order
In this paper, we introduce and investigate two new subclasses of analytic functions in the open unit disk in the complex plane. Several interesting properties of the functions belonging to these classes are examined. Here, sufficient, and necessary and sufficient, conditions for the functions belonging to these classes, respectively, are also given. | Turk J Math (2018) 42: 2423 – 2435 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Some subclasses of analytic functions of complex order Nizami MUSTAFA∗ Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we introduce and investigate two new subclasses of analytic functions in the open unit disk in the complex plane. Several interesting properties of the functions belonging to these classes are examined. Here, sufficient, and necessary and sufficient, conditions for the functions belonging to these classes, respectively, are also given. Furthermore, various properties like order of starlikeness and radius of convexity of the subclasses of these classes and radii of starlikeness and convexity of these subclasses are examined. Key words: Analytic function, coefficient bound, starlike function, convex function 1. Introduction and preliminaries Let A be the class of analytic functions f (z) in the open unit disk U = {z ∈ C : |z| α, z ∈ U , α ∈ [0, 1) , ( ) } zf ′′ (z) f ∈ S : Re 1 + ′ > α, z ∈ U , α ∈ [0, 1) . f (z) () () ∗Correspondence: nizamimustafa@ 2010 AMS Mathematics Subject Classification: 30C45, 30C50, 30C55 2423 This work is licensed under a Creative Commons Attribution International License. MUSTAFA/Turk J Math For convenience, S ∗ = S ∗ (0) and C = C(0) are, respectively, starlike and convex functions in U . It is easy to verify that C ⊂ S ∗ ⊂ S . For details on these classes, one could refer to the monograph by Goodman [5]. Note that we will use T S ∗ (α) = T ∩ S ∗ (α), T C(α) = T ∩ C(α), and in the special case we have T S ∗ = T ∩ S ∗ , T C = T ∩ C for α = 0 . An interesting unification of the function classes S ∗ (α) and C(α) is provided by the class S ∗ C(α, β) of functions f ∈ S , which .
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