关键词: Analytic function, starlike and convex function, differential operator, differ-ential subordination

Abstract:

Let H(U) be the space of analytic functions in the unit disk U. For the integral operator A^Φ,φ_α,β,ν : K → H(U), with K ⊂H(U), defined by
A^Φ,φ_α,β,ν[f](z) =[β+ν/z^νΦ(z)∫^z₀f (t)φ(t)t^δ−1dt]^1/ β,
where α, β, ν, δ ∈C and Φ, φ ∈ H(U), we will determine sufficient conditions on g₁, g₂, α, β and ν, such that
zφ(z)[g₁(z)/z]^α< zφ(z)[ f(z)/z]^α< zφ(z)[g₂(z)/z]^α
implies
zΦ(z)[A^Φ,φ_α,β,ν[g₁](z)/z]^β< zΦ(z)[A^Φ,φ_α,β,ν[f](z)/z]^β< zΦ(z)[A^Φ,φ_α,β,ν[g₂](z)/z]^β
.
The symbol “<” stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, zΦ(z)[A^Φ,φ_α,β,ν[[g₁](z)/z]^β is the largest function and zΦ(z)[A^Φ,φ_α,β,ν[g₂](z)/z]^β the smallest function so that the left-hand side, respectively the right-hand side of the above implications hold, for all f functions satisfying the assumption. We give a particular case of the main result obtained for appropriate choices of functions Φ and φ that also generalizes classic results of the theory of differential subordination and superordination.

Key words: Analytic function, starlike and convex function, differential operator, differ-ential subordination