Cases of Equality for Refinements of Bernstein’s Inequality

We identify extremal polynomials for a class of known refinements of Bernstein’s polynomial inequality. For example, the inequality $$\mid zp^\prime (z) - p(z) + p(0)\mid +\mid p(0)\mid \le (n-1) \max_{\mid u\mid \le 1} \mid p(u)\mid,\qquad \mid z\mid \le 1,$$ holds for any polynomial p of degree n ≥ 2, and equality holds for some z, ¦z¦ = 1, if and only if p is a monomial of degree n or else p ≢ 0.