An Application of the Krein-Milman Theorem to Bernstein and Markov Inequalities

Given a trinomial of the form $p(x)=ax^m+bx^n+c$ with $a,b,c\in{\mathbb R}$, we obtain, explicitly, the best possible constant $\mathcal{M}_{m,n}(x)$ in the inequality $$|p'(x)| \le \mathcal{M}_{m,n}(x) \cdot \|p\|,$$ where $x\in[-1,1]$ is fixed and $\|p\|$ is the sup norm of $p$ over $[-1,1]$. This answers a question to an old problem, first studied by Markov, for a large family of trinomials. We obtain the mappings $\mathcal{M}_{m,n}(x)$ by means of classical convex analysis techniques, in particular, using the Krein-Milman approach.