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

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 $$\mathrm{Re}\left(1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right)>-\frac{1}{2}.$$ We show that every section $s_{2n-1}(z)=z+{\sum }_{k=2}^n{a}_{2k-1}{z}^{2k-1}$, 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.

12 pages, 3 figures