近于凸族中奇函数部分和的凸半径
Radius of convexity of partial sums of odd functions in the close-to-convex family
摘要：
我们考虑一类解析和局部单叶函数$f$的形式$f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$，$|z|<1$，满足条件$$ {\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime (z)}\right)>-\frac{1}{2}. $$我们证明了每一节$s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$，$f$，是凸的磁盘$|z|<\sqrt{2}/3$。我们还证明了半径$\sqrt{2}/3$是最好的可能，即数量$\sqrt{2}/3$不能被一个更大的。
英文摘要：
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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