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.


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