## 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.)

### Keywords:

functions with positive real parts; locally univalent
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