Kargar, R., Ebadian, A. (2017). Ozaki's conditions for general integral operator. Sahand Communications in Mathematical Analysis, 05(1), 61-67. doi: 10.22130/scma.2017.17786
Rahim Kargar; Ali Ebadian. "Ozaki's conditions for general integral operator". Sahand Communications in Mathematical Analysis, 05, 1, 2017, 61-67. doi: 10.22130/scma.2017.17786
Kargar, R., Ebadian, A. (2017). 'Ozaki's conditions for general integral operator', Sahand Communications in Mathematical Analysis, 05(1), pp. 61-67. doi: 10.22130/scma.2017.17786
Kargar, R., Ebadian, A. Ozaki's conditions for general integral operator. Sahand Communications in Mathematical Analysis, 2017; 05(1): 61-67. doi: 10.22130/scma.2017.17786
Department of Mathematics, Payame Noor University, I. R. of Iran.
Abstract
Assume that $\mathbb{D}$ is the open unit disk. Applying Ozaki's conditions, we consider two classes of locally univalent, which denote by $\mathcal{G}(\alpha)$ and $\mathcal{F}(\mu)$ as follows \begin{equation*} \mathcal{G}(\alpha):=\left\{f\in \mathcal{A}:\mathfrak{Re}\left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) <1+\frac{\alpha }{2},\quad 0<\alpha\leq1\right\}, \end{equation*} and \begin{equation*} \mathcal{F}(\alpha):=\left\{f\in \mathcal{A}:\mathfrak{Re}\left( 1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\right) >\frac{1 }{2}-\mu,\quad -1/2<\mu\leq 1\right\}, \end{equation*} respectively, where $z \in \mathbb{D}$. In this paper, we study the mapping properties of this classes under general integral operator. We also, obtain some conditions for integral operator to be convex or starlike function.
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