Refining and reversing the weighted arithmetic-geometric mean inequality involving convex functionals and application for the functional entropy
The scalar means have interpretations in convex functional analysis. For a convex functional \(f:\mathscr{H}\to\mathbb{R}\cup\{+\infty\}\) on a Hilbert space \(\mathscr{H}\), the conjugate of \(f\) is defined by \(f^*(y)=\sup_{x\in\mathscr{H}}\mathfrak{Re}\langle y,x\rangle-f(x)\). For every pair \((f,g)\) of convex functionals with \(f^{**}=f\) and \(g^{**}=g\) and every \(\lambda\in(0,1)\), the weighted functional means are defined by \begin{align*} &\mbox{\small{weighted functional arithmetic mean}} \quad \ \mathcal{A}_\lambda(f,g)=(1-\lambda)f+\lambda g;\\ & \mbox{\small{weighted functional harmonic mean}} \quad \ \mathcal{H}_\lambda(f,g)=\left((1-\lambda)f^*+\lambda g^*\right)^*;\\ &\mbox{\small{weighted functional geometric mean}} \quad \mathcal{G}_\lambda(f,g)=\frac{\sin \lambda\pi}{\pi}\int_{0}^{1}\frac{t^{\lambda-1}}{(1-t)^\lambda}\mathcal{H}_t(f,g)dt. \end{align*} An arithmetic-geometric-harmonic means type inequality holds as \[\mathcal{H}_\lambda(f,g)\leq \mathcal{G}_\lambda(f,g)\leq \mathcal{A}_\lambda(f,g).\] In this paper, the authors present some inequalities involving weighted functional means.
scientific article; zbMATH DE number 7460867
scientific article