Sandwich-type theorems for a class of integral operators

Abstract

Let $H(\mathrm{U})$ be the space of all analytic functions in the unit disk $\mathrm{U}$. For a given function $h\in\mathcal{A}$ we define the integral operator $\mathrm{I}_{h;\beta}:\mathcal{K}\rightarrow H(\mathrm{U})$, with $\mathcal K\subset H(\mathrm{U})$, by $$\mathrm{I}_{h;\beta}[f](z)=\left[\beta \int_0^zf^\beta(t)h^{-1}(t)h'(t)\operatorname{d}t\right]^{1/\beta},$$ where $\beta\in\mathbb{C}$ and all powers are the principal ones. We will determine sufficient conditions on $g_1$, $g_2$ and $\beta$ such that $$\left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}g_1(z)\prec \left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}f(z)\prec \left[\frac{zh'(z)}{h(z)}\right]^{1/\beta}g_2(z)$$ implies $$\mathrm{I}_{h;\beta}[g_1](z)\prec\mathrm{I}_{h;\beta}[f](z)\prec \mathrm{I}_{h;\beta}[g_2](z),$$ where the symbol ``$\prec$'' stands for subordination. We will call such a kind of result a {\em sandwich-type theorem}. In addition, $\displaystyle\mathrm{I}_{h;\beta}[g_1]$ will be the {\em largest} function and $\displaystyle\mathrm{I}_{h;\beta}[g_2]$ the {\em smallest} function so that the left-hand side, respectively the right-hand side of the above implication hold, for all $f$ functions satisfying the differential subordination, respectively the differential superordination of the assumption. We will give some particular cases of the main result obtained for appropriate choices of the $h$, that also generalize classic results of the theory of differential subordination and superordination. The concept of differential superordination was introduced by S. S. Miller and P. T. Mocanu like a dual problem of differential subordination