WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE


Doron S. Lubinsky




We prove weighted Markov-Bernstein inequalities of the form $$ ıt_{-ıty}^{ıfty}|f'(x)|^pw(x)\,dx eq C(igma+1)^pıt_{-ıty}^{ıfty}|f(x)|^pw(x)\,dx $$ 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.