Weighted Markov-Bernstein inequalities for entire functions of exponential type (Q2815263)

The main goal of the paper is to prove weighted Markov-Bernstein inequalities of the form NEWLINE\[NEWLINE \int_{-\infty}^{\infty}| f'(x)| ^pw(x)\,dx \leq C(\sigma+1)^p\int_{-\infty}^{\infty}| f(x)| ^pw(x)\,dx NEWLINE\]NEWLINE where the weight \(w\) satisfies certain doubling type properties, \(f\) is an entire function of exponential type \(\leq\sigma\), \(p>0\), and \(C\) is independent of \(f\) and \(\sigma\).NEWLINENEWLINE If the weight satisfies the doubling condition, related result is obtained by \textit{G. Mastroianni} and \textit{V. Totik} [Constructive Approximation Vol. 16, No. 1, 37--71 (2000; Zbl 0956.42001)]. for \(1\leq p<\infty\), and extended to \(0<p<1\) by \textit{T. Erdelyi} [J. Approximation Theory, Vol.100, No. 1, 60--72 (1999; Zbl 0985.41009)]. The weighted Markov-Bernstein inequalities are here proved under the new conditions on the weight \(w\), the one-sided doubling condition around \(0\) and the growth condition about integers. The authors emphasizes the generality of the result by giving several corollaries which are all new results in the form of weighted Markov-Bernstein inequalities. Estimates hold for weights whose choice does not depend on \(\sigma\).