The Bojanov--Naidenov inequality for quartics and second derivatives

Abstract

We settle the case $n=4$, $k=2$ of the Bojanov--Naidenov problem for algebraic polynomials. Let $P$ be a real polynomial of degree at most four with $\left\lVert{P}\right\rVert_{C[-1,1]}\leq 1$, and let $T_4(x)=8x^4-8x^2+1$. We prove that, for every $t\geq0$, \[ \int_{-1}^{1} \bigl(|P''(x)|-t\bigr)_+\,dx \leq \int_{-1}^{1} \bigl(|T_4''(x)|-t\bigr)_+\,dx . \] This tail estimate implies \[ \int_{-1}^{1}φ(|P''(x)|)\,dx \leq \int_{-1}^{1}φ(|T_4''(x)|)\,dx \] for every nondecreasing convex function $φ:[0,\infty)\to\mathbb{R}$. If $φ$ is strictly increasing and convex, equality can occur only for $P=\pm T_4$. The proof is elementary and finite. We interpolate at the five extremal points of $T_4$; convexity then reduces the problem to the $32$ sign choices at these nodes. At each vertex the second derivative is a quadratic polynomial, so the remaining work is an explicit comparison of level sets.