On some intermediate mean values

We give a necessary and sufficient mean condition for the quotient of two Jensen functionals and define a new class ${\mathrm{\Lambda }}_{f,g}(a,b)$ of mean values where $f,g$ are continuously differentiable convex functions satisfying the relation $f\mathrm{"}(t)=tg\mathrm{"}(t),t\in {\mathbb{R}}^{+}$. Then we asked for a characterization of $f,g$ such that the inequalities $H(a, b)\le \Lambda_{f, g}(a, b)\le A(a, b)$or$L(a, b)\le \Lambda_{f, g}(a, b)\le I(a, b)$holdforeachpositive$a, b$,where$H, A, L, I$ are the harmonic, arithmetic, logarithmic and identric means, respectively. For a subclass of $\mathrm{\Lambda }$ with $g\mathrm{"}(t)=t^s,s\in \mathbb{R}$, this problem is thoroughly solved.