Büheri和Tammi公式中的最优控制
Optimal control in Bombieri's and Tammi's conjectures
摘要：
令$S$代表单位圆盘$U=\{z: |z|<1\}$中由$U$中的$f(z)=z+a_2z^2+...$规范化的单叶正则函数的通常类，并令$S^M$是通过限制$U$中的$|f(z)|<M$而定义的其子类，$M\geq 1$。我们考虑两个经典问题：一类$S$的系数问题和由$S^M$得到的函数第四系数的精确估计.使用Löwner的参数表示和最优控制方法，我们给出了确切的初始Bjueri的号码，并推导出一个尖锐的常数$M_0$，使所有的$M\geq M_0$的挑选功能给出局部最大值$|a_4|$。给出了数值近似。
英文摘要：
Let $S$ stand for the usual class of univalent regular functions in the unit disk $U=\{z: |z|<1\}$ normalized by $f(z)=z+a_2z^2+...$ in $U$, and let $S^M$ be its subclass defined by restricting $|f(z)|<M$ in $U$, $M\geq 1$. We consider two classical problems: Bombieri's coefficient problem for the class $S$ and the sharp estimate of the fourth coefficient of a function from $S^M$. Using Löwner's parametric representation and the optimal control method we give exact initial Bombieri's numbers and derive a sharp constant $M_0$, such that for all $M\geq M_0$ the Pick function gives the local maximum to $|a_4|$. Numerical approximation is given.
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