The Laub–Ilani inequality [‘A subtle inequality’, Amer. Math. Monthly97 (1990), 65–67] states that $x^{x}+y^{y}\geqslant x^{y}+y^{x}$ for nonnegative real numbers $x,y$. We introduce and prove new trigonometric and algebraic-trigonometric inequalities of Laub–Ilani type and propose some conjectural algebraic-trigonometric inequalities of similar forms.

A classical theorem due to Young states that the cosine polynomial

$$ C_n(x)=1+\sum_{k=1}^{n}\frac{\cos(kx)}{k} $$

is positive for all $n\geq1$ and $x\in(0,\pi)$. We prove the following refinement. For all $n\geq2$ and $x\in[0,\pi]$ we have

$$ \tfrac{1}{6}+c(\pi-x)^2\leq C_n(x), $$

with the best possible constant factor

$$ c=\min_{0\leq t\lt\pi}\frac{5+6\cos(t)+3\cos(2t)}{6(\pi-t)^2}=0.069\dots. $$

Let pn(x) be a real algebraic polynomial of degree n, and consider the Lp norms on I = [−1, 1]. A classical result of A. A. Markoff states that if ‖pn‖. ∞ ≤ 1, then ‖P′n‖∞ ≤ n2. A generalization of Markoff's problem, first suggested by P. Turán, is to find upper bounds for ‖pn(J)‖p if ∣pn(x)∣≤ ψ(x)x ∈ I. Here ψ(x) is a given function, a curved majorant. In this paper we study extremal properties of ‖p′n‖2 and ‖p″n‖2 if pn(x) has the parabolic majorant ∣p(x)∣≤ 1 − x2, x ∈ I. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ‖(1 − x2)