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Zbl 1211.30032
Tuneski, Nikola
Convex functions and functions with bounded turning.
(English)
[J] Tamsui Oxf. J. Math. Sci. 26, No. 2, 161-172 (2010). ISSN 1561-8307

Summary: Let $\Cal A$ be the class of analytic functions in the unit disk $\Cal U=\{z:\vert z\vert<1\}$ that are normalized by $f(0)=f'(0)-1=0$, and let $-1\le B<A\le 1$ and $-1\le D<C\le 1$. In this paper, the following generalizations of the class of convex functions and of the class of functions with bounded turning are studied: $$K[A,B]=\bigg\{f\in\Cal A: 1+\frac{zf''(z)}{f'(z)}\prec\frac{1+Az}{1+Bz}\bigg\}$$ and $$R_k^{}[C,D]=\bigg\{f\in\Cal A: \big(f'(z)\big)^{1/k}\prec\frac{1+Cz}{1+Cz}\bigg\},\qquad k\ge 1.$$Conditions for $K[A,B]\subset R_k[D,D]$ are given, together with several corollaries for different choices of $A$, $B$, $C$, $D$, and $k$,.
MSC 2000:
*30C45 Special classes of univalent and multivalent functions

Keywords: convex function, differential subordination

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