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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.

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