ON CERTAIN INTEGRAL TRANSFORMATIONS
Shigeyoshi Owa · Institutional Repositories DataBase (IRDB) · 1995
The object of the present paper is to derive some subordination properties of certain integral transformations of functions which are analytic in the open unit disk. $1\mathrm{N}\mathrm{T}\mathrm{R}0\mathrm{D}\mathrm{u}\mathrm{C}\mathrm{T}$ IONLet A be the class of functions of the form $\mathrm{f}(\mathrm{z})$ $=\mathrm{z}+$ $\mathrm{n}=2t\mathrm{a}_{\mathrm{n}}\mathrm{z}^{\mathrm{n}}$ which are analyCic in the open unit disk $\mathrm{U}=\{\mathrm{z}: |\mathrm{z}| <1\}$ .For func tions $\mathrm{f}(\mathrm{z}\rangle$ $\in$ A and $\mathrm{g}(\mathrm{z})$ $\in \mathrm{A}$ , we say that $\mathrm{f}(\mathrm{z})$ is subordinate to $\mathrm{g}(\mathrm{z})$ if there exis Cs an analyCic func Cion $\mathrm{w}(\mathrm{z})$ i.n $\mathrm{U}$ which satisfies $\mathrm{w}(0)$ $=0$ , $|\mathrm{w}(\mathrm{z})|$ $<$ $1$ $(\mathrm{z}\in \mathrm{U})$ .and $\mathrm{f}(\mathrm{z})=\mathrm{g}(\mathrm{w}(\mathrm{z}))$ .We denote this subordination by $\mathrm{f}(\mathrm{z})\prec \mathrm{g}(\mathrm{z})$ .If $\mathrm{g}(\mathrm{z})$ is univalent in $\mathrm{U}$ , Chen this subordination $\mathrm{f}(\mathrm{z})\prec \mathrm{g}(\mathrm{z})$ is equivalent to $\mathrm{f}(0)$ $=\mathrm{g}(0)$ and $\mathrm{f}(\mathrm{u})\mathrm{C}\mathrm{g}(\mathrm{u})$ .For a functi.on$\mathrm{f}(\mathrm{z})$ belonging to $\mathrm{A}$ , we define the following integral Cransformation I $(\mathrm{f}(\mathrm{z}))$ by I $(\mathrm{f}(\mathrm{z}))$ $= \{\frac{\alpha+8}{\mathrm{z}^{8}}\int_{\mathrm{o}^{\mathrm{t}^{8-1_{\mathrm{f}}}}}^{\mathrm{Z}}(\mathrm{t})\mathrm{a}_{\mathrm{d}\mathrm{t}\}}$ $(\mathrm{z}\in \mathrm{u})$ , where $\alpha\in \mathrm{C}$ , $\mathrm{a} eq 0$ , and 8 $\in \mathrm{C}$ .To derive some subordination $\acute{\mathrm{p}}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s}$ of the integral cransformations I $(\mathrm{f}(\mathrm{z}))$ , we have to recall here the $\mathrm{f}$ ollowing lemmas.1991 Ma Chematic $\mathrm{s}$ Subj ect Class ification.Pr imary $30\mathrm{C}45$ .