Inequalities of the Rado-Popoviciu type for functions and their applications
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.
scientific article; zbMATH DE number 3859441
scientific article