APPLICATIONS OF DIFFERENTIAL SUBORDINATIONS TO CERTAIN CLASSES OF STARLIKE FUNCTIONS

Abstract

Let p be an analytic function defined on the open unit disk D. We obtain certain differential subordination implications such as ψ(p) := pλ(z)(α+βp(z)+γ/p(z)+δzp′(z)/pj(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to P, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy | log(zf ′(z)/f(z))| < 1, |(zf ′(z)/f(z))2 − 1| < 1 and zf ′(z)/f(z) lying in the parabolic region v2 < 2u− 1.