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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 $\vert z\vert<1$; here $0\leq\rho<1$, $k\geq 2$: $P\sb k(\rho)$ of functions $p(z)$ with $p(0)=1$ and, $z=re\sp{i\theta}$, $$\int\sb 0\sp{2\pi}\left\vert{{{\cal R}p(z)-\rho} \over {1- \rho}}\right\vert d\theta\leq k\pi$$ (for $\rho=0$, $k=2$, this gives the family $P$ of functions with positive real parts); $V\sb k(\rho)$ of functions $f(z)$, locally univalent, with $f(0)=0$, $f'(0)=1$ and $(zf'(z))'(f'(z))\sp{-1}\in P\sb k(\rho)$; $T\sb k(\rho)$ of functions $f(z)$ with $f(0)=0$, $f'(0)=1$, such that there exists $g\in V\sb k(\rho)$ with $f'(z)(g'(z))\sp{-1}\in P$. He proves in a straightforward manner various results for these families, in particular indicating their relationship to other special families of functions.

30C45Special classes of univalent and multivalent functions
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