On Some Intermediate Mean Values

Slavko Simic Mathematical Institute SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia Correspondence should be addressed to Slavko Simic; [email protected] Received 25 June 2012; Revised 9 December 2012; Accepted 16 December 2012 Academic Editor: Mowaffaq Hajja Copyright © 2013 Slavko Simic.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class Λ f,g(a, b) of mean values where f, g are continuously differentiable convex functions satisfying the relation f(t) = tg(t), t ∈ R. Then we asked for a characterization of f, g such that the inequalitiesH(a, b) ≤ Λ f,g(a, b) ≤ A(a, b) or L(a, b) ≤ Λ f,g(a, b) ≤ I(a, b) hold for each positive a, b, whereH,A, L, I are the harmonic, arithmetic, logarithmic, and identric means, respectively. For a subclass of Λ with g(t) = t, s ∈ R, this problem is thoroughly solved.