Inequalities involving Dresher variance mean

Abstract

Let p be a real density function defined on a compact subset of R m , and let E(f , p) = pf dω be the expectation of f with respect to the density function p. In this paper, we define a one-parameter extensionof a positive continuous function f defined on . By means of this extension, a two-parameter mean V r,s (f , p), called the Dresher variance mean, is then defined. Their properties are then discussed. In particular, we establish a Dresher variance mean inequality min t∈ {f (t)} ≤ V r,s (f , p) ≤ max t∈ {f (t)}, that is to say, the Dresher variance mean V r,s (f , p) is a true mean of f . We also establish a Dresher-type inequality V r,s (f , p) ≥ V r * ,s * (f , p) under appropriate conditions on r, s, r * , s * ; and finally, acan be compared with E(f , p). We are also able to illustrate the uses of these results in space science. MSC: 26D15; 26E60; 62J10

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