论文标题
纽曼类型限制为$ l_p [-1,1] $ - 多项式的对数衍生物的均值
A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle
论文作者
论文摘要
令$ g_n $，$ n = 1,2，\点$，是一个复杂多项式的对数导数，在单位圆上所有零，即$ g_n（z）=（z-z-z_ {1}）的函数， $ | z_1 | = \ dots = | z_n | = 1 $。对于任何$ p> 0 $，我们建立了bound \ [\ int _ { - 1}^1 | g_n（x）|^p \，dx> c_p \，n^{p-1}，\ \]在数量$ n $的顺序上，在$ c_p> 0 $的情况下，仅在$ c_p>的情况下，仅在$ p $ p $上。特定情况$ p = 1 $的这种不平等现象可以被视为众所周知的估计$ \ iint_ {| z | <1} | g_n（z）| \，dxdy> c> 0 $的$ g_n $集成区域的$ c> 0 $，由d.jj.获得。纽曼（1972）。结果还表明，在空间$ l_p [-1,1] $，$ p \ ge 1 $中，集合$ \ {g_n \} $并不密度。
Let $g_n$, $n=1,2,\dots$, be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $g_n(z)=(z-z_{1})^{-1}+\dots+(z-z_{n})^{-1}$, $|z_1|=\dots=|z_n|=1$. For any $p>0$, we establish the bound \[\int_{-1}^1 |g_n(x)|^p\, dx>C_p\, n^{p-1},\] sharp in the order of the quantity $n$, where $C_p>0$ is a constant, depending only on $p$. The particular case $p=1$ of this inequality can be considered as a stronger variant of the well-known estimate $\iint_{|z|<1} |g_n(z)|\,dxdy>c>0$ for the area integral of $g_n$, obtained by D.J. Newman (1972). The result also shows that the set $\{g_n\}$ is not dense in the spaces $L_p[-1,1]$, $p\ge 1$.
