Inequality for fractional integrals of complex polynomials in \(L_0\). (English. Russian original) Zbl 1314.30007

Storozhenko, É. A.; Kovalenko, L. G.

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

Keywords:

complex polynomial; fractional integral; Bernstein inequality; Favard inequality

References:

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