On polynomials monotonic on the unit interval
The reviewed paper contains the proof of the following theorem 1. Let \(f\) be a polynomial of even degree \(n\) with \(|f(x)|\leq 1\) for \(x\in [-1,1]\). Besides, let \(f\) be monotonic on \([-1,1]\). Then NEWLINE\[NEWLINE |f^\prime(x)|\leq \tfrac{1}{4}n(n+2),\quad x\in [-1,1]\tag{\(*\)} NEWLINE\]NEWLINE The proof is a technical one, but rather complicated. The representation formula for monotonic polynomials is used. The extremal polynomial is determined and is given via Jacobi polynomials \(P^{(0,1)}(x).\) The corresponding bound in \((*)\) for odd degree polynomials was known earlier and is equal \(\frac{1}{4}(n+1)^2.\)
scientific article; zbMATH DE number 1647682