A unified approach to radius of convexity problems for certain classes of univalent analytic functions
Let P(\(\alpha\),\(\beta)\) denote the class of functions p analytic in the unit disk E, with \(p(0)=1\) and satisfying \[ | (p(z)-1)/2\beta (p(z)- \alpha)-(p(z)-1)| <1\quad for\quad some\quad \alpha,\beta \] (0\(\leq \alpha <1\), \(0\leq \beta <1)\) and all \(z\in E\), and let \[ P_ n(\alpha,\beta)=\{p:\quad p(z)=1+b_ nz^ n+b_{n+1}z^{n+1}+...,\quad p\in P(\alpha,\beta)\}. \] Also denote by \(R_ n(\alpha,\beta)\) the class of functions f of the form \(f(z)=z+a_{n+1}z^{n+1}+...\) analytic in E, such that f'\(\in P(\alpha,\beta)\). It has been shown earlier that \[ | a_{n+1}| \leq 2\beta (1-\alpha)/(n+1). \] If \[ P_ n(b;\alpha,\beta)=\{p\in P_ n(\alpha,\beta):\quad b_ n=2\beta (1-\alpha)b,\quad 0\leq b\leq 1\} \] and \[ R_ n(a;\alpha,\beta)=\{f(z)=z+(2\beta (1- \alpha)/(n+1))az^{n+1}+...:\quad f'\in P_ n(a;\alpha,\beta),\quad 0\leq a<1\}, \] the authors determine the sharp bound for the radius of convexity of the class \(R_ n(a;\alpha,\beta)\). Analogously the class \(S^*_ n(\alpha,\beta)\) is defined as the class of functions of the form \(g(z)=z+a_{n+1}z^{n+1}+...\) analytic in E such that zg'(z)/g(z)\(\in P(\alpha,\beta)\), \(z\in E\), and \[ S^*_ n(a;\alpha,\beta)=\{g(z)=z+(2\beta (1-\alpha)/n)z^{n+1}+...:\quad zg'/g\in P_ n(a;\alpha,\beta),\quad 0\leq a\leq 1\}. \] A sharp estimate for the radius of convexity for \(S^*_ n(a;\alpha,\beta)\) is obtained. Several known results are deduced.
scientific article; zbMATH DE number 3910717