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On subclasses of close-to-convex functions of higher order. (English) Zbl 0758.30010

The author introduces several classes of functions regular in \(| z|<1\); here \(0\leq\rho<1\), \(k\geq 2\):
\(P_ k(\rho)\) of functions \(p(z)\) with \(p(0)=1\) and, \(z=re^{i\theta}\), \[ \int_ 0^{2\pi}\left|{{{\mathcal R}p(z)-\rho} \over {1- \rho}}\right| d\theta\leq k\pi \] (for \(\rho=0\), \(k=2\), this gives the family \(P\) of functions with positive real parts);
\(V_ k(\rho)\) of functions \(f(z)\), locally univalent, with \(f(0)=0\), \(f'(0)=1\) and \((zf'(z))'(f'(z))^{-1}\in P_ k(\rho)\);
\(T_ k(\rho)\) of functions \(f(z)\) with \(f(0)=0\), \(f'(0)=1\), such that there exists \(g\in V_ k(\rho)\) with \(f'(z)(g'(z))^{-1}\in P\).
He proves in a straightforward manner various results for these families, in particular indicating their relationship to other special families of functions.

MSC:

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