Original scientific paper

A class of superordination-preserving convex integral operator

Saibah Siregar ; School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, 43 600 Bangi, Selangor, Malaysia
Maslina Darus ; School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, 43 600 Bangi, Selangor, Malaysia
Teodor Bulboacă ; Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400 084 Cluj-Napoca, Romania

Abstract

If $H(\mathrm{U})$ denotes the space of analytic functions in the unit disk $\mathrm{U}$, for the integral operator $A_{\alpha,\beta,\gamma,\delta}^h:\mathcal K\rightarrow H(\mathrm{U})$, with $\mathcal K\subset H(\mathrm{U})$, defined by
\[
A_{\alpha,\beta,\gamma,\delta}^h[f](z)=\left[\frac{\beta+\gamma}{z^\gamma}
\int_0^zf^\alpha(t)h(t)t^{\delta-1}\operatorname{d}t\right]^{1/\beta},\;
\Big(\alpha,\beta,\gamma,\delta\in\mathbb{C}\;\text{and}\;\;h\in H(\mathrm{U})\Big),
\]
we will determine 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
\[
z\left[\frac{A_{\alpha,\beta,\gamma,\delta}^h[g_1](z)}{z}\right]^\beta\prec
z\left[\frac{A_{\alpha,\beta,\gamma,\delta}^h[f](z)}{z}\right]^\beta\prec
z\left[\frac{A_{\alpha,\beta,\gamma,\delta}^h[g_2](z)}{z}\right]^\beta.
\]

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 has been held for all $f$ functions satisfying the double differential subordination of the assumption.

The results generalize those of the last author from \cite{bul3}, obtained for the special case $\alpha=\beta$ and $h\equiv1$.

Keywords

analytic function; starlike and convex function; differential operator; differential subordination

Hrčak ID:

44030

URI

https://hrcak.srce.hr/44030

Publication date:

9.12.2009.

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