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On a subclass of certain starlike functions with negative coefficients. II. (English) Zbl 0867.30016
Summary: For part I see O. Altintas in Math. Jap. 36, No. 3, 489-495 (1991; Zbl 0739.30011).
Let \(P(n,\lambda,\alpha)\) denote the class of functions \(f(z)=z-\sum_{k=n+ 1}^\infty a_kz^k\) \((a_k\geq 0)\) which are analytic in the unit disc \(U=\{z:|z|<1\}\) and satisfy the conditions \(\text{Re} \left\{{zf'(z)+ \lambda z^2f''(z) \over\lambda zf'(z)+ (1-\lambda) f(z)}\right\} >\alpha\) for some \(\alpha (0\leq\alpha<1)\), \(\lambda (0\leq\lambda\leq 1)\) and for all \(z \in U\). In this paper we obtain closure theorems, and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class \(P(n,\lambda,\alpha)\). Also modified Hadamard products of several functions belonging to the class \(P(n,\lambda,\alpha)\) are studied. Furthermore, some interesting distortion inequalities for certain fractional integral operators are shown.
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)