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On harmonic univalent functions defined by a generalized Ruscheweyh derivatives operator. (English) Zbl 1114.30012
Summary: Let \({\mathcal S}_{\mathcal H}\) denote the class of functions \(f=h+ \overline g\) which are harmonic univalent and sense preserving in the unit disk \(U\). Al-Shaqsi and Darus [On univalent functions with respect to \(K\)-symmetric points given by a generalised Ruscheweyh derivatives operator (to appear)] introduced a generalized Ruscheweyh derivatives operator denoted by \(D^n_\lambda\) where \(D^n_\lambda f(z)=z+\sum^\infty_{k=2}[1+\lambda(k-1)]C(n,k) a_k z^k\), where \(C(n,k)={k+n-1\choose n}\). The authors, using this operators, introduce the class \({\mathcal H}^n_\lambda\) of functions which are harmonic in \(U\). Coefficient bounds, distortion bounds and extreme points are obtained.

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