New extension of Alexander and Libera integral operators

Authors: HATUN ÖZLEM GÜNEY, SHIGEYOSHI OWA

Abstract: Let $T$ be the class of analytic functions in the open unit disc $\mathbb{U}$ with $f(0)=0$ and $f'(0)=1.$ For $f(z)\in T,$ the Alexander integral operator $A_{-1}f(z),$ the Libera integral operator $L_{-1}f(z)$ and the Bernardi integral operator $B_{-1}f(z)$ were considered before. Using $A_{-1}f(z)$ and $L_{-1}f(z),$ a new integral operator $F_{\lambda}f(z)$ is considered. After discuss some properties of dominant for $F_{\lambda}f(z),$ another new integral operator $O_{-1}f(z)$ of $f(z)\in T$ is discussed. The object of the present paper is to discuss the dominant of new integral operators $F_{\lambda}f(z)$ and $O_{-1}f(z)$ concerning with some starlike functions and convex functions in $\mathbb{U}.$

Keywords: Analytic function, Alexander integral operator, Libera integral operator, Bernardi integral operator, dominant, new integral operator

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