DIFFERENTIAL SUBORDINATIONS AND α-CONVEX FUNCTIONS

This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = { z : | z | < 1 } satisfying the conditions that f （0） = 0, f’（0） = 1, and α(1 + （zf’’（z））/f’（z） + （1-α） （zf’（z））/f（z） ∈P（U）for all z ∈ U , where α is a real number and P（z） = （1 + τ2z2）/（1-τz-τ2z2） （z ∈ U ）.The number τ = （1 -5（1/2））/2 is such that τ2 = 1 + τ . The class SLMα introduced by J. Dziok, R.K. Raina, and J. Sokó [3, Appl. Math. Comput. 218 （2011）, 996 1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.