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A class of double subordination-preserving integral operators. (English) Zbl 1108.30017

Summary: If $H\left(U\right)$ denotes the space of analytic functions in the unit disk $U$, for the function $h\in 𝒜$ and $\beta \in ℂ$, we define the integral operator ${I}_{h;\beta }:𝒦\subset H\left(U\right)\to H\left(U\right)$ by

${I}_{\beta ,\gamma }\left(f\right)\left(z\right)={\left[\beta {\int }_{0}^{z}{f}^{\beta }\left(t\right){h}^{-1}\left(t\right){h}^{\text{'}}\left(t\right)dt\right]}^{1/\beta }·$

If “$\prec$” stands for subordination, we determine simple sufficient conditions on ${g}_{1}$, ${g}_{2}$ and $\beta$ such that

${\left[\frac{z{h}^{\text{'}}\left(z\right)}{h\left(z\right)}\right]}^{1/\beta }{g}_{1}\left(z\right)\prec {\left[\frac{z{h}^{\text{'}}\left(z\right)}{h\left(z\right)}\right]}^{1/\beta }f\left(z\right)\prec {\left[\frac{z{h}^{\text{'}}\left(z\right)}{h\left(z\right)}\right]}^{1/\beta }{g}_{2}\left(z\right)$

implies

${I}_{h;\beta }\left[{g}_{1}\right]\left(z\right)\prec {I}_{h;\beta }\left[f\right]\left(z\right)\prec {I}_{h;\beta }\left[{g}_{2}\right]\left(z\right),$

and say that ${I}_{h;\beta }$ is a double subordination-preserving integral operator, and we call such a kind of result a sandwich-type theorem.

Moreover, we prove that the above implication is sharp, in the sense that ${I}_{h;\beta }\left[{g}_{1}\right]$ is the largest function and ${I}_{h;\beta }\left[{g}_{2}\right]$ the smallest function so that the left-hand side, respectively the right-hand side of the above implication hold.

MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions