A Newman type bound for \(L_p[-1,1]\)-means of the logarithmic derivative of polynomials having all zeros on the unit Circle (Q6182694)

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 \(g_n(z)=(z-z_0)^{-1}+\cdots + (z-z_n)^{-1}\), \(|z_1|=\cdots =|z_n |=1\). It is proved that for the function \(g_n\) and \(p>0\) the inequality \[ \int_{-1}^1 |g_n(x)|^pdx> \int_{-1}^1 |g_n(x)|^p|x|^pdx > C_pn^{p-1} \] is correct, where \(C_p>0\) is a constant, depending only on \(p\) and the set \(\{g_n\}\) is not dense in the spaces \(L_p[-1,1], p\neq 1\). Besides, in the cases \(p=1\) and \(p=2\), this equality is true for the constant \(C_1=1/50\) and \(C_2=1/213\). The obtained result generalizes one well-known D. Newman's result.