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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.
##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)