On a problem of Turán about polynomials with curved majorants

Abstract:

Let $\phi (x) \geqq 0$ for $- 1 \leqq x \leqq 1$. For a fixed ${x_0}$ in $[ - 1,1]$ what can be said for $\max |{p’_n}({x_0})|$ if ${p_n}(x)$ belongs to the class ${P_\phi }$ of all polynomials of degree $n$ satisfying the inequality $|{p_n}(x)| \leqq \phi (x)$ for $- 1 \leqq x \leqq 1$? The case $\phi (x) = 1$ was considered by A. A. Markov and S. N. Bernstein. We investigate the problem when $\phi (x) = {(1 - {x^2})^{1/2}}$. We also study the case $\phi (x) = |x|$ and the subclass consisting of polynomials typically real in $|z| < 1$.