一个积分算子的单叶性

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国家教育委员会博士点基金资助项目

详细信息

ON THE UNIVALENCE OF AN INTEGRAL OPERATOR

摘要: 引入了一个定义在单位圆 $\mathcal{U}=\{z\in\mathbb{C}:|z|1 \}$ 内规范化的解析函数类 $\mathscr{A}$ 上的积分算子 $J_{\gamma_1,\cdots,\gamma_n,\beta}(z)$ , 利用著名的Becker单叶性判别法, Schwarz引理和Caratheodory不等式, 得到了这个积分算子在单位圆内单叶的3个充分条件. 即当 $f_{j}(z)(j=1,2,\cdots,n)$ 及参数 $\gamma_{1},\cdots,\gamma_{n},\beta$ 满足一定条件时, 积分算子 $J_{\gamma_1,\cdots,\gamma_n,\beta}(z)$ 在单位圆内是单叶的.
- 解析函数 /
- 积分算子 /
- 单叶性 /
- 星象性
Abstract: A general integral operator $J_{\gamma_1,\cdots,\gamma_n,\beta}(z)$ is introduced, which is defined on the class ~ $\mathscr{A}$ of normalized analytic functions in ~ $\mathcal{U}=\{z\in\mathbb{C}:|z|1 \}$ . Three sufficient conditions for the univalence of this integral operator in the unit disk $\mathcal{U}$ are provided by applying the well-known Becker univalence criteria, Schwarz lemma and Caratheodory inequality. That is, the integral operator $J_{\gamma_1,\cdots,\gamma_n,\beta}(z)$ is univalent in the unit disk $\mathcal{U}$ when the functions $f_{j}(z)(j=1,2,\cdots,n)$ and the parameters $\gamma_{1},\cdots,\gamma_{n},\beta$ satisfy some conditions.
- analytic functions /
- integral operator /
- univalence /
- starlike property