Chebyshev Inequality in Function Spaces

Abstract

We study the classical Chebyshev inequality

$${\int }_0^{a}x(s)ds{\int }_0^{a}y(s)ds\le a{\int }_0^{a}x(s)y(s)ds$$

and prove new variants, generalizations and abstractions. Among these are Chebyshev’s inequality for strongly increasing functions, positive convex and concave functions, and generalizations of the Ky Fan inequality. Our abstractions involve Chebyshev’s inequality in Banach function spaces and symmetric spaces. These considerations lead to generalizations of inequalities of Favard and Barnes. The constants that arise are studied and shown to be sharp in general.