On Some Intermediate Mean Values. (11th February 2013) - Simic, Slavko
Abstract : 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 ) = t g ″ ( t ), t ∈ ℝ + . Then we asked for a characterization of f, g such that the inequalities H ( 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, where H, A, L, I are the harmonic, arithmetic, logarithmic, and identric means, respectively. For a subclass of Λ with g ″ ( t ) = t s, s ∈ ℝ, this problem is thoroughly solved.