On certain subclasses of close-to-convex functions (Q1112973)

Let \(A_ n\) denote the class of regular functions such that \[ f(z)=z+\sum^{\infty}_{k=n+1}a_ kz^ k\quad (n=1,2,...)\quad (| z| <1). \] Moreover, let \(R_ n\) denote the subclass of \(A_ n\) satisfying the condition \[ | f'(z)-1| <1-\alpha \quad (0\leq \alpha <1). \] Hence the authors remark the close-to-convexity of \(R_ n\). Next the authors prove the following result by means of \textit{S. S. Miller} and \textit{P. T. Mocanu}'s lemma [J. Math. Anal. Appl. 65, 289-305 (1978; Zbl 0367.34005)]. Let \(f(z)\in R_ n\) and \(\prec\) denote subordination. Then \[ f(z)/z\quad \prec \quad 1+(1-\alpha)z/(n+1)\quad (n=1,2,...). \]