A Subclass of Uniformly Convex Functions Associated with Certain Fractional Calculus Operator


In this paper, we introduce a new class Kµ,,�(�,�) of uniformly convex functions defined by a certain fractional calculus operator. The class has interesting subclasses like �-uniformly starlike, �- uniformly convex and �-uniformly pre-starlike func- tions. Properties like coefficient estimates, growth and distortion theorems, modified Hadamard prod- uct, inclusion property, extreme points, closure the- orem and other properties of this class are studied. Lastly, we discuss a class preserving integral oper- ator, radius of starlikeness, convexity and close-to- convexity and integral mean inequality for functions in the class Kµ,,�(�,�). Notice that, UST(α,0) = S(α) and UCV (α,0) = K(α), where S(α) and K(α) are respectively the popular classes of starlike and convex functions of order α (0 ≤ α < 1). The classes UST(α, β) and UCV (α, β) were introduced and studied by Goodman (4), Rønning (13) and Minda and Ma (8). Clearly f ∈ UCV (α, β) if and only if zf ' ∈ UST(α, β).A function f(z) is said to be close-to-convex of order r, 0 ≤ r < 1 if Ref '(z) > r. Let φ(a, c; z) be the incomplete beta function defined by

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