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Integral inequalities for algebraic polynomial on the unit circle. (Russian) Zbl 0713.30006
This paper is a sequel of a previous work of the author on polynomials [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 3-22 (1981; Zbl 0538.42001)]. In that paper the following inequality was established: Theorem A. For an arbitrary polynomial of degree \(nP_ n\) and an arbitrary polynomial \(\Lambda_ n\) with zeros only in the closed interior or the closed exterior of the unit disk the sharp inequality \[ H_{\phi}(\Lambda_ nP_ n)\leq H_{\phi}(\alpha (\Delta_ n)P_ n),\quad p>0 \] holds. Here \(\Lambda(z)= \Lambda_ n(z)= \sum^{n}_{k=0} \binom{n}{k} \lambda_ kz^ k\), \(P_ n(z)= \sum^{n}_{k=0} \binom{n}{k} a_ kz^ k\), \(\Lambda P_ n(z)= \sum^{n}_{k=0} \binom{n}{k} \lambda_ ka_ kz^ k\). \(H_{\phi}(P)= \int^{2\pi}_{0} \phi(| P(e^{it})|)dt\). Furthermore \(\alpha (\Lambda_ n)= \max(| \lambda_ 0|\), \(| \lambda_ n|)\). The function \(\phi\) is defined on the positive real axis, increasing and absolutely continuous and such that \(x\phi '(x)\) is increasing. Define the norms \[ \| f\|_ p= \left( \frac{1}{2\pi}\int^{2\pi}_{0}| f| (e^{it})|^ pdt \right)^{1/p},\quad 0<p<\infty, \] \[ \| f\|_ 0=\lim_{p\to +0}\| f\|_ p. \] The author generalizes Theorem A in Theorem 1 \[ H_{\phi}(\Lambda_ nP_ n)\leq H_{\phi}(\| \Lambda_ n\|_ 0P_ n). \] The author also identifies all the extremal polynomials. For \(\Lambda\) with zeros in the closed unit disk denote by \(\nu_ n(\Lambda)_ p\) the p-norm of \(\Lambda\) in the space of polyomials of degree n. The author shows, as a corollary of the previous theorem: Corollary \[ \nu (\Lambda)_ 2\leq \nu_ n(\Lambda)_ p\leq \nu_ n(\Lambda)_ 0. \] In Theorem 2 the author proves the converse of Theorem 1. provided the constant \(\| \Lambda_ n\|_ 0\) in Theorem 1 is the smallest possible for the function \(\phi(u)=u^ 2.\)
The author also considers the problem of best constants in the inequalities: \[ \| P_ n+Q_ n\|_ 0\leq \chi(n)(\| P_ n\|_ 0+\| Q_ n\|_ 0) \] in the space of polynomials of degree n.The author proves the inequality \[ \frac{1}{2}\rho^ n\leq \chi(n)\leq \frac{1}{2}R^ n \] for \(n\geq 6\). Here \(\rho =\exp (\frac{2G}{\pi})= 1.7916...G= \sum^{\infty}_{\nu =0}\frac{(- 1)^{\nu}}{(2\nu +1)^ 2}= 0.9159...\). and \(R=^ 6\sqrt{60}= 1.8493..\).
Reviewer: Z.Rubinstein

30C10 Polynomials and rational functions of one complex variable
26D05 Inequalities for trigonometric functions and polynomials
26D15 Inequalities for sums, series and integrals