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- W4318763453 abstract "Let $$g_n$$ , $$n=1,2,ldots $$ , be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form $$g_n(z)=(z-z_{1})^{-1}+cdots +(z-z_{n})^{-1}$$ , $$|z_1|=cdots =|z_n|=1$$ . For any $$p>0$$ , we establish the bound $$begin{aligned} int _{-1}^1 |g_n(x)|^p, dx>C_p, n^{p-1}, end{aligned}$$ 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 Newman (Am Math Mon 79(9):1015–1016, 1972). The result also shows that the set $${g_n}$$ is not dense in the spaces $$L_p[-1,1]$$ , $$pge 1$$ ." @default.
- W4318763453 created "2023-02-02" @default.
- W4318763453 creator A5065927496 @default.
- W4318763453 date "2023-02-01" @default.
- W4318763453 modified "2023-10-12" @default.
- W4318763453 title "A Newman type bound for $$L_p[-1,1]$$-means of the logarithmic derivative of polynomials having all zeros on the unit circle" @default.
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- W4318763453 doi "https://doi.org/10.1007/s00365-023-09622-8" @default.
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