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Zbl 1182.30022
Siregar, Saibah; Darus, Maslina; Bulboacă, Teodor
A class of superordination-preserving convex integral operator. (English)
[J] Math. Commun. 14, No. 2, 379-390 (2009). ISSN 1331-0623

Summary: If $H(U)$ denotes the space of analytic functions in the unit disk $U$, for the integral operator $A^h_{\alpha,\beta,\gamma,\delta}:\Cal K\to H(U)$, with $\Cal K\subset H(U)$, defined by $$ A^h_{\alpha,\beta,\gamma,\delta}[f](z)=\left[\frac{\beta+\gamma}{z^\gamma}\int_0^z f^\alpha(t)h(t)t^{\delta-1} dt\right],\quad\alpha,\beta,\gamma,\delta\in \mathbb C\text{ and }h\in H(U), $$ we determeine sufficient conditions on $g_1$, $ g_2$, $\alpha$, $\beta$, and $\gamma$ such that $$ zh(z)\left[\frac{g_1(z)}{z}\right]^\alpha \prec zh(z)\left[\frac{f(z)}{z}\right]^\alpha \prec zh(z)\left[\frac{g_2(z)}{z}\right]^\alpha $$ implies $$ z\left[\frac{A^h_{\alpha,\beta,\gamma,\delta}[g_1](z)}{z}\right]^\beta \prec z\left[\frac{A^h_{\alpha,\beta,\gamma,\delta}[f](z)}{z}\right]^\beta \prec z\left[\frac{A^h_{\alpha,\beta,\gamma,\delta}[g_2](z)}{z}\right]^\beta . $$ In addition, both of the subordinations are sharp, since the left-hand side will be the largest function, and the right-hand side will be the smallest function so that the above implication holds for all functions $f$ satisfying the double differential subordination of the assumption. The results generalize those of the last author [{\it T. Bulboac}, Bull. Korean Math. Soc. 34, No.4, 627--636 (1997; Zbl 0898.30021)], obtained for the special case $\alpha=\beta$ and $h\equiv 1$.

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: analytic functions in the unit disc; differential subordination

Citations: Zbl 0898.30021

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