Some inequalities for polynomials with a prescribed zero

Rahman, Q. I.; Schmeisser, G.

doi:10.1090/S0002-9947-1976-0399427-7

Some inequalities for polynomials with a prescribed zero
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by Q. I. Rahman and G. Schmeisser PDF

Trans. Amer. Math. Soc. 216 (1976), 91-103 Request permission

Abstract:

Let $f(z)$ be a polynomial of degree n. Given that $f(z)$ has a zero on the circle $|z| = \rho \;(0 < \rho < \infty )$, we estimate $|f(0)|$ and \[ {({(2\pi )^{ - 1}}\int _0^{2\pi }|f({e^{i\theta }}){|^2}d\theta )^{1/2}}\] in terms of ${\max _{|z| = 1}}|f(z)|$. We also consider some other related problems.

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