Properties of certain integral operator (Q916819)

Let \(A_ n(\alpha)\) denote the class of analytic functions \(f(z)=\sum^{\infty}_{k=n+1}a_ kz^ k\), \(n\in \{1,2,...\}\), in the unit disk U, which satisfy in \[ U| f(z)/z-1| <1-\alpha,\quad 0\leq \alpha <1, \] and let \(J_ c\) denote the generalized Libera integral operator, defined by \[ J_ c(f)(z)=((c+1)/z^ c)\int^{z}_{0}t^{c-1}f(t)dt,\quad c\geq 0. \] In this note, the authors prove that for \(f\in A_ n(\alpha)\), \[ J_ c(f)(z)/z\quad \prec \quad 1+(1-\alpha)z/(n+1)\text{ and } Re\{e^{i\beta}J_ c(f)(z)/z\}>0 \] for \(| \beta | \leq \pi /2-\sin^{-1}(1- \alpha)/(n+1)\). The bound of \(\beta\) is the best possible.