On Quotients of L^p-Means

Abstract

In this paper we propose a method for establishing upper bounds for functions of the form

$$\Phi (f) = \frac{{{{\left( {\int {{f^p}d\mu } } \right)}^{1/p}}}}{{{{\left( {\int {{f^q}d\lambda } } \right)}^{1/q}}}}$$

, where μ and λ are probability measures and q is less than p. Specifically, it is shown that, subject to certain conditions, ϕ is a quasiconvex function and therefore satisfies a boundary-maximum principle. This yields a unified explanation of the “vertex phenomenon” in the theory of complementary inequalities (cf. [6], Section 2). The lower bound for ϕ is also determined.