论文标题
关于类$ {\ mathcal u}（λ）$的第五系数的猜想
On a conjecture for the fifth coefficients for the class ${\mathcal U}(λ)$
论文作者
论文摘要
让$ f $为单位磁盘$ {\ mathbb d} = \ {z：| z | | <1 \} $的函数，使得$ f（0）= f'（0）-1 = 0 $，I.E.如果另外，\ [\ left | \ left（\ frac {z} {f（z）} \ right）^2 f'（z）-1 \ right |<λ\ quad \ quad \ quad \ quad（z \ in {\ mathbb d}），然后$ f $属于class $ {\ mathcal u}（mathcal u}（λ）$，$ 0 <re1 $。在本文中，我们证明了从$ {\ mathcal u}（λ）$满足\ [\ frac {f（z）} {z} {z} {z} \ prec \ prec \ frac {1} {（1+z）（1+λz）（1+λz）（us frac）的第五系数的$ f $的尖锐上限。 $ 0.400436 \ ldots \leλ\ le1 $。
Let $f$ be function that is analytic in the unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$, i.e., of type $f(z)=z+\sum_{n=2}^{\infty} a_n z^n$. If additionally, \[ \left| \left(\frac{z}{f(z)}\right)^2 f'(z) -1\right|<λ\quad\quad (z\in{\mathbb D}), \] then $f$ belongs to the class ${\mathcal U}(λ)$, $0<λ\le1$. In this paper we prove sharp upper bound of the modulus of the fifth coefficient of $f$ from ${\mathcal U}(λ)$ satisfying \[ \frac{f(z)}{z}\prec \frac{1}{(1+z)(1+λz)}, \] ("$\prec$" is the usual subordination) in the case when $0.400436\ldots \leλ\le1$.
