On the identric and logarithmic means

Abstract

Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b b /a a ) 1/(b−a) , fora ≠ b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b − a)/(logb − loga), fora ≠ b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) =\(\sqrt {ab}\). In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.