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Zbl 1108.30017
Bulboacă, Teodor
A class of double subordination-preserving integral operators. (English)
[J] PU.M.A., Pure Math. Appl. 15, No. 2-3, 87-106 (2004). ISSN 1218-4586

Summary: If $H(U)$ denotes the space of analytic functions in the unit disk $U$, for the function $h\in\Cal{A}$ and $\beta \in \Bbb{C}$, we define the integral operator $\operatorname{I}_{h;\beta}:\Cal{K}\subset H(U)\rightarrow H(U)$ by $$\operatorname{I}_{\beta,\gamma}(f)(z)=\left[\beta \int_0^zf^\beta(t)h^{-1}(t)h'(t)\operatorname{d}t\right]^{1/\beta}.$$ If ``$\prec$'' stands for subordination, we determine simple 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 $$\operatorname{I}_{h;\beta}[g_1](z)\prec\operatorname{I}_{h;\beta}[f](z)\prec \operatorname{I}_{h;\beta}[g_2](z),$$ and say that $\operatorname{I}_{h;\beta}$ 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 $\displaystyle\operatorname{I}_{h;\beta}[g_1]$ is the largest function and $\displaystyle\operatorname{I}_{h;\beta}[g_2]$ the smallest function so that the left-hand side, respectively the right-hand side of the above implication hold.

MSC 2000:: *30C80 Maximum principle, etc. (one complex variable)
30C45 Special classes of univalent and multivalent functions

Keywords: integral operators; subordination-preserving

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