SOME RESULTS ASSOCIATED WITH CERTAIN ANALYTIC AND UNIVALENT FUNCTIONS INVOLVING FRACTIONAL DERIVATIVE OPERATORS

Abstract. This paper investigates some results (Theorems 2.1-2.3, below) concerning certain classes of analytic and univalentfunctions, involving the familiar fractional derivative operators.We state interesting consequences arising from the main resultsby mentioning the cases connected with the starlikeness, convex-ity, close-to-convexity and quasi-convexity of geometric functiontheory. Relevant connections with known results are also empha-sized briefly. 1. Introduction and DefinitionsLet A n denote the class of functions f(z) normalized byf(z) = z +a n+1 z n+1 +a n+2 z n+2 +...(n ∈ N ≡ {1,2,3,··· }), (1.1)which are analytic and univalent in the open unit disc U = {z ∈C : |z| < 1}.We denote by S ∗n (α), K n (α), C n (α,β) and C ∗n (α,β), the subclassesof the class A n consisting of functions which are, respectively, starlikeof order α, convex of order α, close-to-convex of order β and type α,and quasi-convex of order β and type α in U, where 0 ≤ α < 1 and Received September 20, 2005.2000 Mathematics Subject Classification: 30C45, 26A33, 30A10.Key words and phrases: Normalized analytic functions, univalent functions,fractional calculus operators, convexity, starlikeness, close-to-convexity, quasi-convexity, inequalities, Jack’s Lemma, Nunokawa’s Lemma.