Mixed means and inequalities of Hardy and Levin-Cochran-Lee type for multidimensional balls

Abstract

Abstract. Integral means of arbitrary order, with power weights and their companion means, where the integrals are taken over balls in R n centered at the origin, are introduced and related mixed-means inequalities are derived. These relations are then used in obtaining Hardy and Levin-Cochran-Lee inequalities and their companion results for n-dimensional balls. Finally, the best possible constants for these inequalities are obtained.

Multidimensional Exponential Inequalities with Weights

“…Inequality (1.2) has also been investigated by many authors (see [2,4,7,8,[10][11][12][13][15][16][17][19][20][21]23,24] and the references therein). The higher dimensional theory of (1.1) and (1.2) for k(x, t) = α|S x | −α |S t | α−1 is discussed by Heinig [9], Drábek-Heinig-Kufner [5], Jain-Persson-Wedestig [14] for α = 1 and byČižmešija-Pečarić-Perić [3] for α > 0. In these papers, E = R n , S x is the ball B(|x|) in R n centered at the origin and of radius |x|, and |S x | is the volume of S x .…”