WEIGHTED MARKOV–BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

Doron S. Lubinsky

Abstract: We prove weighted Markov–Bernstein inequalities of the form $$ \int_{-\ifty}^{\infty}|f'(x)|^pw(x) dx \leq C(\sigma+1)^p\int_{-\ifty}^{\infty}|f(x)|^pw(x) dx $$ Here $w$ satisfies certain doubling type properties, $f$ is an entire function of exponential type $\leq\sigma$, $p>0$, and $C$ is independent of $f$ and $\sigma$. For example, $w(x)=(1+x^2)^{\alpha}$ satisfies the conditions for any $\alpha\in\mathbb{R}$. Classical doubling inequalities of Mastroianni and Totik inspired this result.

Keywords: entire functions of exponential type, Bernstein inequalities

Classification (MSC2000): 42C05

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