Inequalities of the Rado-Popoviciu type for functions and their applications
type: Item
instance of: scholarly article
title: Inequalities of the Rado-Popoviciu type for functions and their applications
zbMATH Open document ID: 0541.26009
DOI: 10.1016/0022-247X(84)90092-1
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: The main result of this paper is a simple proof of the following extremely general inequalities: let \(\lambda,\mu,\nu \in R, \lambda \mu>0\), \(\nu \geq 0\), f:\(R\to R\), \(f>0\) and multiplicatively convex, a,p,q positive n-tuples \(P_ k=\sum^{k}_{i=1}p_ i,\) \[ A_ n(f)=(1/Q_ n)\sum^{n}_{i=1}q_ if(a_ i),\quad f(A_ n)=f[(1/P_ n)\sum^{n}_{i=1}p_ ia_ i]: \] then (a) if \(\mu q_ n-Q_ n<0\) \[ P_ nA_ n(f)-\lambda(p_ n/q_ n)Q_ nf(A_ n)^{\mu}\geq \] \[ \geq P_{n-1}A_{n-1}(f)-(\lambda \mu)^{Q_ n/(Q_ n-\mu q_ n)}f(A_{n-1})^{\mu Q_{n-1}/(Q_ n-\mu q_ n)}(Q_ n/\mu -q_ n); \] \[ (b)\quad \frac{[A_ n(f)+\nu]^{P_ n}}{[f(A_ n)]^{p_ nQ_ n/q_ n}}\geq \frac{[A_{n-1}(f)+\nu P_ n/P_{n-1}]^{P_{n- 1}}}{[f(A_{n-1})]^{p_ nQ_{n-1}/q_ n}}; \] equality occurs iff \(Q_{n-1}a_ n=\sum^{n-1}_{i=1}p_ ia_ i\) and, in (a), \[ f(a_ n)=(\lambda \mu)^{Q_ n/Q_ n-\mu q_ n}f(A_{n-1})^{\mu Q_{n- 1}/Q-\mu q_ n}, \] in (b), \(f(a_ n)=A_ n(f)+\nu\). These results include inequalities of a similar nature due to Mitrinović and Vasić and if \(\mu =\lambda =1, \nu =0\), \(p=q\), \(f=\exp\) reduce to the classical results of Rado and Popoviciu.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 3859441
zbMATH Keywords: Rado-Popoviciu type inequalities
zbMATH Keywords: arithmetic-geometric mean inequality
zbMATH Keywords: Jensen inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3859441