Comparison of means generated by two functions and a measure
Let \(I\subset{\mathbb R}\) be an open interval, \(f,g:I\to{\mathbb R}\) be continuous, \(g\)~positive and \(f/g\) strictly monotone. Let \(\mu\) be a~probability Borel measure on \([0,1]\). Then \[ M_{f,g;\mu}(x,y):=\biggl(\frac{f}{g}\biggr)^{-1} \left( \frac{\int_0^1f\bigl(tx+(1-t)y\bigr)\,d\mu(t)} {\int_0^1g\bigl(tx+(1-t)y\bigr)\,d\mu(t)} \right),\qquad x,y\in I \] defines a~class of means containing many well known means: Bajraktarević, quasiarithmetic, Cauchy, Lagrangian, Gini, Stolarsky etc. The comparison problem is studied for these means. Conditions for \((f,g)\), \((h,k)\) and measures \(\mu\), \(\nu\) are found that the inequality \(M_{f,g;\mu}(x,y)\leq M_{h,k;\nu}(x,y)\) (\(x,y\in I\)) holds. Necessary conditions and sufficient conditions are studied.
scientific article; zbMATH DE number 5292294
scientific article