  We answer Totik's question on weighted Bernstein's inequalities showing that
$$ \|T_n'\|_{L_p(\omega)} \le C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad
0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain
nondoubling weights $\omega$. Moreover, we find necessary conditions on
$\omega$ for Bernstein's inequality to hold. We also prove weighted
Bernstein-Markov, Remez, and Nikolskii inequalities for trigonometric and
algebraic polynomials.
