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On a subclass of certain starlike functions with negative coefficients. (English) Zbl 0739.30011
The author defines a new class of analytic functions with negative coefficients, $$P(n,\lambda,\alpha)$$, a generalization of classes defined by H. Silverman [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] and H. M. Srivastava, S. Owa and S. K. Chatterjea [Rend. Semin. Mat. Univ. Padova 77, 115-124 (1987; Zbl 0596.30018)]. $f(z)=z-\sum_{k=n+1}^ \infty a_ k t^ k,\qquad a_ k>0,\quad n\in N,$ analytic in the unit disk $$U$$ is said to be in $$P(n,\lambda,\alpha)$$ if it satisfies $\hbox{Re}\{[zf'(z)+\lambda z^ 2f''(z)]/[\lambda zf'(z)+(1-\lambda)f(z)]\}>\alpha$ for some $$\alpha$$, $$0\leq\alpha<1$$, $$\lambda$$, $$0\leq\lambda\leq 1$$ and for $$z\in U$$. Distortion inequalities, the order of starlikeness and results for fractional integral and derivatives are given.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) 26A33 Fractional derivatives and integrals