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Classes of functions defined by certain differential-integral operators. (English) Zbl 0946.30007

Let \({\mathcal A}(p, k)\), \(p,k\in\mathbb{N}\), \(p<k\), denote the class of functions of the form \(f(z)= z^p+ a_k z^k+ a_{k+1} z^{k+1}+\cdots\), which are holomorphic in the unit disc \({\mathcal U}\). Let \(T(\alpha, \beta)\) denote the class of functions \(f\in{\mathcal A}(p, k)\) satisfying the following condition: \(\Omega^\alpha_\beta f(z)/z^p\prec(1+ Az)/(1+ Bz)\), \(z\in{\mathcal U}\). Denote by \(T_\theta(\alpha,\beta)\) the subclass of the class \(T(\alpha,\beta)\) of functions \(f\) such that \(\arg a_n= \theta\) for \(a_n\neq 0\), \(n= k+ 1,k+ 2,\dots\) . The linear operator \(\Omega^\alpha_\beta:{\mathcal A}(p, k)\to{\mathcal A}(p, k)\) is defined here by the known operators \(Q^\alpha_\beta\) and \(\Phi^\alpha_\beta\) from the papers: J. B. Jung, Y. C. Kim, H. M. Srivastava (1993) and Y. C. Kim, K. S. Lee, H. M. Srivastava (1992). In this paper, the author investigates the class \(T_\theta(\alpha, \beta)\). Coefficient estimates, distortion theorems, extreme points, the radii of convexity and starlikeness are given.

MSC:

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

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