A CLASS OF ANALYTIC-FUNCTIONS DEFINED BY FRACTIONAL DERIVATION

Let f(z) = z +... be analytic in the unit disc \z\ < 1 and 0 less than or equal to lambda less than or equal to 1. We define a linear operator by Q(lambda)f = Gamma(2 - lambda)z(lambda)D(z)(lambda)f(z), where D(z)(lambda)f(z) denotes the fractional derivative of f(z) The function f(z) is said to be in R(lambda, alpha) if it satisfies the condition Re{Q(lambda)f/z} > alpha, 0 less than or equal to alpha < 1, \z\ < 1. In this paper, we prove, for 0 less than or equal to u less than or equal to lambda < 1, that R(lambda, alpha) subset of or equal to R(u, alpha) and study some subordination properties. We also obtain distortion theorems and a coefficient inequality for R(lambda, alpha). Finally, we discuss the Hadamard product of the class R(lambda,alpha). (C) 1994 Academic Press, Inc.