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arXiv · 1212.3895

On some intermediate mean values

Slavko Simic
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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 $f"(t)=t g"(t), t\in \Bbb R^+$. Then we asked for a characterization of $f, g$ such that the inequalities $H(a, b)\le Λ_{f, g}(a, b)\le A(a, b)$ or $L(a, b)\le Λ_{f, g}(a, b)\le I(a, b)$ 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 $g"(t)=t^s, s\in \Bbb R$, this problem is thoroughly solved.

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Slavko Simic. 2012-12-17. On some intermediate mean values. https://arxiv.org/abs/1212.3895

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