Subordinations by alpha-convex functions

Abstract

Let H(U) be the space of analytic functions in the unit disk U and let \mathscr{D}={φ∈H(U):φ(0)=1, φ(z)≠0, z∈U}. For the functions φ, φ∈\mathscr{D} we will determine simple sufficient conditions such that
[\frac{φ(z)}{φ(z)+(1/γ)zφ'(z)}]^1/βf(z) {\prec} k(z){⇒}I_{φ, φ;β, γ}[f](z) {\prec} k(z),
for all k∈\mathscr{M}_1/β', where
I_{φ, φ;β, γ}[f](z)=[\frac{γ}{z^γφ(z)}∫₀^zf^β(t)t^γ−1φ(t) dt]^1/β
and \mathscr{M}_1/β' represents the class of 1/β-convex functions (not necessarily normalized).
In particular, we will give sufficient conditions on φ and φ so that the operators I_{φ, φ;β, γ} are averaging operators on certain subsets of H(U). In addition, some particular cases of the main result, obtained for appropriate choices of the φ and φ functions, will also be given.