On Some Intermediate Mean Values

Abstract EN

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class Λf,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 inequalities H(a,b)≤Λf,g(a,b)≤A(a,b) or L(a,b)≤Λf,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)=ts, s∈ℝ, this problem is thoroughly solved.