The objective of the present paper is to determine the sharp radii of convexity for the integral operator$C_{\alpha ,\beta }[f,g](z)=\underset{0}{\overset{z}{\int }}{\left(\frac{f(w)}{w}\right)}^{\alpha }{(g^{\prime }(w))}^{\beta }dw,\alpha ,\beta \in \mathbb{R},|z|<1,$ when f and g range over some classes of classical analytic functions defined in the open unit disk $|z|<1$ with $f(0)=0$. Here, the exponents are chosen in such a way that the respective functions are analytic in suitable branches. As an importance of considering the operator $C_{\alpha ,\beta }$, we observe the radii of convexity in some classical situations through several consequences of our main results.

中文翻译：

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涉及Hornich运算的积分算子的凸半径

本文的目的是为积分算子确定凸的尖锐半径$C_{\alpha ，\beta }[F，G]（ž）=\underset{0}{\overset{ž}{\int }}{（\frac{F（w）}{w}）}^{\alpha }{（G^{\prime }（w））}^{\beta }dw，\alpha ，\beta \in \mathbb{[R}，|ž|<\mathrm{1个}，$当f和g超出开放单位磁盘中定义的某些经典分析函数的范围时$|ž|<\mathrm{1个}$ 和 $F（0）=0$。在此，选择指数的方式应使各个函数在适当的分支中进行分析。作为考虑运营商的重要性$C_{\alpha ，\beta }$，我们通过主要结果的一些结果来观察某些经典情况下的凸半径。