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Inequality for fractional integrals of complex polynomials in $$L_0$$. (English. Russian original) Zbl 1314.30007
Math. Notes 96, No. 4, 609-612 (2014); translation from Mat. Zametki 96, No. 4, 633-636 (2014).
From the text: “Let
$P_n(z)=\sum_{k=0}^nc_kz^k$
be an algebraic polynomial of degree $$n$$ with complex coefficients. For $$0\leq p\leq \infty$$, the functionals $$\|\cdot\|_p^{}$$ on the unit circle $$| z|=1$$ are defined, as usual, by the equalities
$\| P_n\|_\infty=\max| P_n(z)|,\quad\| P_n\|_p=\bigg(\frac{1}{2\pi}\int_0^{2\pi}| P_n(e^{i\varphi})^p d\varphi\bigg)^{1/p},$
$\| P_n\|_0=\lim_{p\to\infty}\| P_n\|_p=\exp\bigg(\frac{1}{2\pi}\int_0^{2\pi}| P_n(e^{i\varphi})^p d\varphi\bigg)^{1/p}.$
$$[\ldots]$$ the quasinorm $$\| P_n\|_0$$ will be called the measure of the polynomial $$P_n(z)$$. $$[\ldots]$$ In the present paper, we consider fractional integrals of order $$0<\alpha<1$$ and prove the analog of inequality (2’) for measures of polynomials. $$[\ldots]$$”

##### MSC:
 30C10 Polynomials and rational functions of one complex variable
Full Text:
##### References:
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