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.
Categories: math.CV