Subordinations by alpha-convex functions

Abstract

Let $H(\mathrm{U})$ be the space of analytic functions in the unit disk $\mathrm{U}$ and let $\mathcal{D}=\{\varphi\in H(\mathrm{U}):\varphi(0)=1, \varphi(z)\neq0,z\in\mathrm{U}\}$. For the functions $\phi,\varphi\in\mathcal{D}$ we will determine simple sufficient conditions such that $$\left[\frac{\varphi(z)}{\phi(z)+\frac{1}{\gamma}z\phi'(z)}\right]^{1/\beta} f(z)\prec k(z)\Rightarrow\operatorname{I}_{\phi,\varphi;\beta,\gamma}[f](z) \prec k(z),$$ for all $k\in\mathcal{M'}_{1/\beta}$, where $$\operatorname{I}_{\phi,\varphi;\beta,\gamma}[f](z)= \left[\frac{\gamma}{z^\gamma\phi(z)} \int_0^zf^\beta(t)t^{\gamma-1}\varphi(t) \operatorname{d}t\right]^{1/\beta}$$ and $\mathcal{M'}_{1/\beta}$ represents the class of $1/\beta$-convex functions (not necessarily normalized). In particular, we will give sufficient conditions on $\phi$ and $\varphi$ so that the operators $\operatorname{I}_{\phi,\varphi;\beta,\gamma}$ are averaging operators on certain subsets of $H(\mathrm{U})$. In addition, some particular cases of the main result, obtained for appropriate choices of the $\phi$ and $\varphi$ functions, will also be given.