Abstract
Using an integral inequality, contributions are made towards the solutions of two long open problems. The first one concerns the determination of the best constant $C(p)$ in Bernstein’s inequality
\[
\int_0^{2\pi} |H_n'(t)|^p dt \leqq C(p)n^p \int_0^{2\pi} |H_n(t)|^p dt, \qquad 0 < p < 1,
\]
where $H_n$ is a trigonometric polynomial of degree $n$. We prove $C(p)\leqq 11$, while previously $C(p) \leqq 8/p$ was known.
The second one concerns orthogonal polynomials $p_n(d\alpha)$ corresponding to positive measures $d\alpha$ defined on $[-1,1]$. We prove that
\[
\mathrm{lim\; sup}_{n\to \infty} n^{-1} {\textstyle\sum}_{k=\alpha}^n\; p_k^2(d\alpha,x) < \infty \qquad \mathrm{a. e.}\; \mathrm{in}\; [-1,1]
\]
provided that $d\alpha$ belongs to the Szegö class, i.e., if
\[
\int_{-1}^1 |\log \alpha'(t)| \sqrt{1-t^2} dt < \infty.
\]
These results have interesting applications in approximation theory, probability, and statistics; e.g., the second one is known to imply that if $d\alpha$ belongs to the Szegö class and $f\in L_{d^n}^2$ then the orthogonal series of $f$ in $p_n(d\alpha)$ is summable $(C,\varepsilon)$ to $f$ almost everywhre for every $\varepsilon > 0$.
