where and denote the arithmetic and geometric means of with weights , . We prove:
(i) The inequality
is valid for all and with if and only if .
(ii) Inequality with “” instead of “” holds for all and with if and only if and .
This extends a result of Dragomir, Comănescu and Pearce, who proved for the special case , .
"INEQUALITIES FOR WEIGHTED ARITHMETIC AND GEOMETRIC MEANS." Real Anal. Exchange 46 (2) 359 - 366, 2021. https://doi.org/10.14321/realanalexch.46.2.0359