A study on the generalization of Janowski functions in the unit disc

Abstract

Let › be the class of functions w(z), w(0) = 0, |w(z)| < 1 regular in the unit disc D = {z : |z| < 1}. For arbitrarily fixed numbers A 2 (¡1,1), B 2 (¡1,A), 0 • fi < 1 let P(A,B,fi) be the class of regular functions p(z) in D such that p(0) = 1, and which is p(z) 2 P(A,B,fi) if and only if p(z) = 1+((1¡fi)A+fiB)w(z) 1+Bw(z) for some function w(z) 2 › and every z 2 D. In the present paper we apply the principle of subordination ((1), (3), (4), (5)) to give new proofs for some classical results concerning the class S⁄(A,B,fi) of functions f(z) with f(0) = 0, f0(0) = 1, which are regular in D satisfying the condition: f(z) 2 S⁄(A,B,fi) if and only if z f 0 (z) f(z) = p(z) for some p(z) 2 P(A,B,fi) and for all z in D.