On integral operators of certain subclasses of Carathéodory functions (Q2771168)

Let \(\mathcal A\) be the class of all analytic functions \(f\) in the unit disc \(\{z: |z|<1\}\) such that \(f(0)=f'(0)-1=0\). For \(0<\alpha\leq 1\), define \(R^{*}(\alpha)=\{f\in {\mathcal A}: |f'(z)-1|<\alpha\}\). The first main result of this paper is that if \(f \in R^{*}(\alpha)\) then \({1+c \over z^{c}}\int_{0}^{z}t^{c-1}f(t)dt \in R^{*}(\beta)\), \(\beta ={\alpha(c+1)\over c+2}\). The sharp version of a more general result is already known in the literature (for instance one can refer the proof of Corollary 2 in the reviewer's paper [Proc. Indian Acad. Sci. No.~2, 397-411 (1994; Zbl 0808.30012)]. Another result (Theorem 2.2) is also known in the same paper.