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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

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