论文标题
对数系数的求解解决方案
Solution of the logarithmic coefficients conjecture in some families of univalent functions
论文作者
论文摘要
对于单价和归一化的功能$ f $对数系数$γ_n（f）$由公式$ \ log（f（z）/z）= \ sum_ {n = 1}^{\ infty} {\ infty}2γ_n（f）2γ_n（f）z^n $确定。在论文\ cite {pon}中，作者提出了一个猜想，即单位磁盘中的本地函数，满足条件\ [\ [\ re \ left \ left \ {1+zf''（z）/f'（z）/f'（z）\ right \ right \ right \ right \ right \ right \ right \ right \ right \ 1+λ/2+λ/2 \ quad（z \ quad（z \ quad in \ in \ int \ nr \ nathb}} d} 还满足以下不等式：$$ |γ_n（f）| \leλ/（2n（n+1））。$$这里$λ$是一个真实数字，因此$ 0 <λ\ le 1 $。在论文中，我们确认猜想是真实而敏锐的。
For univalent and normalized functions $f$ the logarithmic coefficients $γ_n(f)$ are determined by the formula $\log(f(z)/z)=\sum_{n=1}^{\infty}2γ_n(f)z^n$. In the paper \cite{Pon} the authors posed the conjecture that a locally univalent function in the unit disk, satisfying the condition \[ \Re\left\{1+zf''(z)/f'(z)\right\}<1+λ/2\quad (z\in \mathbb{D}), \] fulfill also the following inequality: $$|γ_n(f)|\le λ/(2n(n+1)).$$ Here $λ$ is a real number such that $0<λ\le 1$. In the paper we confirm that the conjecture is true, and sharp.
