General Mathematics, Vol. 4, 1996

Eugen Draghici -- A general class of starlike integral operators defined on the class of functions with bounded argument rotation

Abstract: Let ${\cal U}$ be the open complex unit disc and let ${\cal A}$ be the class of all functions $f$, analytic in ${\cal U}$ and normed with the conditions $f(0)=0$, $f'(0)=1$. Let ${\cal B}$ be the class of all analytic functions $h$, defined on ${\cal U}$ and with $h(0)=1$. If $\gamma$ is a real number, greated then $-1$ and $g\in {\cal A}$, $h\in {\cal B}$, we define the integral operator\\ $I_{g,h}:{\cal A}\rightarrow {\cal A}$ by $F(z)={\cal I}(f)(z)$, where: \[ F(z)=\frac{\gamma +1}{g^{\gamma}(z)}\int\limits_{0}^{z}f(u)g^{\gamma -1}(u)h(u)du\ \ \ ,\ \ \ z\in {\cal U}\ . \] If ${\cal R}=\{f\in {\cal A}:Re\, f'(z)>0,z\in {\cal U}\}$ is the class of functions with bounded rotation in ${\cal U}$ and $S^{\ast}=\displaystyle{\left\{f\in {\cal A}:\frac{Re\, zf'(z)}{f(z)}>0,z\in {\cal U}\right\}}$ is the class of starlike functions in ${\cal U}$, we will obtain sufficient conditions on $\gamma ,g$ and $h$ so that ${\cal I}_{g,h}({\cal R})\subset S^{\ast}$.

Classification (MSC91): 30C42

Keywords: bounded rotation, convexity of order alpha, starlike function, subordination.

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