Estimates on the Derivative of a Polynomial with a Curved Majorant Using Convex Techniques

A mapping $\phi\colon [-1,1]\rightarrow [0,\infty)$ is a curved majorant for a polynomial $p$ in one real variable if $|p(x)|\leq \phi(x)$ for all $x\in[-1,1]$. If ${\mathcal P}_n^\phi({\mathbb R})$ is the set of all one real variable polynomials of degree at most $n$ having the curved majorant $\phi$, then we study the problem of determining, explicitly, the best possible constant $\mathcal{M}^\phi_{n}(x)$ in the inequality $$|p'(x)| \le \mathcal{M}^\phi_n(x)\|p\|,$$ for each fixed $x\in[-1,1]$, where $p\in {\mathcal P}_n^\phi ({\mathbb R})$ and $\|p\|$ is the sup norm of $p$ over the interval $[-1,1]$. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely $\phi(x) = \sqrt{1-x^2}$ or $\phi(x) = |x|$ for all $x\in[-1,1]$ (circular and linear majorant respectively), were first studied by Q. I. Rahman [``On a problem of Tur{\'a}n about polynomials with curved majorants'', Trans. Amer. Math. Soc. 163 (1972) 447--455]. In that reference the author provided, for each $n\in{\mathbb N}$, the maximum of $\mathcal{M}^\phi_n(x)$ over $[-1,1]$ as well as an upper bound for $\mathcal{M}^\phi_n(x)$ for each $x\in[-1,1]$, where $\phi$ is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach)