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Harmonic functions starlike of the complex order. (English) Zbl 1121.30012

Let \(TS_H\) denote the family of functions \(f=h+\bar{g}\) that are harmonic in the unit disc \(U\) with normalization \(h(z)=z-\sum_{n=2}^\infty a_nz^n\), \(g(z)=\sum_{n=1}^\infty b_nz^n\), \(a_n,b_n\geq0\), \(b_1<1\). The authors consider the subclass \(TS_H^\star(\gamma)\) of \(TS_H\) consisting of functions \(f=h+\bar{g}\in TS_H\) that satisfy the condition \[ \operatorname{Re}\left\{1+\frac1\gamma\biggl(\frac{zh'(z)-\overline{zg'(z)}}{h(z)+\overline{g(z)}}-1\biggr)\right\}>0, \quad \gamma\in C\smallsetminus\{0\}. \] They give necessary and sufficient conditions for functions to be in \(TS_H^\star(\gamma)\).

MSC:

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