There is a well known fact, that if we have non-decreasing weight $w(x)$ in the finite interval $[a,b]$, then if $\{ p_n(x) \}$ is the set of corresponding orthogonal polynomials, the functions $|\sqrt{w(x)}p_n(x)|$ attain their maximum in $[a,b]$ for $x=b$. (the proof can be found in Szego book)
My question is: are there any additional properties about the growth of polynomials, if, for example, we know, that $w(x)$ is convex/concave ($w’’(x) \geq 0$ or $w’’(x) \leq 0$)?