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.


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