A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle

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 $g_n(z)=(z-z_1{)}^{-1}+\cdots +(z-z_n{)}^{-1}$, $\mathrm{\mid }{z}_1\mathrm{\mid }=\cdots =\mathrm{\mid }{z}_n\mathrm{\mid }=1$. For any $p>0$, we establish the bound $${\int }_{-1}^1\mathrm{\mid }{g}_n(x){\mathrm{\mid }}^{p}dx>C_p{n}^{p-1},$$ 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 ${\iint }_{\mathrm{\mid }z\mathrm{\mid }<1}\mathrm{\mid }{g}_n(z)\mathrm{\mid }dxdy>c>0$ 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 $L_p[-1,1]$, $p\ge 1$.