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

By Sarita Agrawal, S. Sahoo. 2016

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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| -\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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7fd1658b401e9feb49b38521c3bfcfef4cdad7af