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 + τ2z21 − τz − τ2z2 (z ∈ u).The number τ = (1 − 5)/2 is such that τ2 = 1 + τ. The class SLMα introduced by J. Dziok, R.K. Raina, and J. Sokół [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.