Radius of convexity of partial sums of odd functions in the close-to-convex family

Sarita Agrawal; Swadesh Kumar Sahoo

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arXiv · 1604.05500

Radius of convexity of partial sums of odd functions in the close-to-convex family

Sarita Agrawal·Swadesh Kumar Sahoo

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Abstract

We consider the class of all analytic and locally univalent functions $f$ of the form $f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$, $|z|<1$, satisfying the condition $$ {\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime (z)}\right)>-\frac{1}{2}. $$ We show that every section $s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$, of $f$, is convex in the disk $|z|<\sqrt{2}/3$. We also prove that the radius $\sqrt{2}/3$ is best possible, i.e. the number $\sqrt{2}/3$ cannot be replaced by a larger one.

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BibTeX RIS

Sarita Agrawal, Swadesh Kumar Sahoo. 2016-04-19. Radius of convexity of partial sums of odd functions in the close-to-convex family. https://arxiv.org/abs/1604.05500

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