A Variance–AM–GM Inequality Conjecture: A Call to the Mathematical Community

This note proposes a new inequality linking the classical arithmetic and geometric means with the variance of positive real variables. The conjectured relation, which we call the Variance–AM–GM Inequality, offers a quantitative bridge between additive and multiplicative dispersion. Its resolution may yield new insights in convex analysis, information theory, and statistical physics. The purpose of this paper is to formally state the conjecture, provide motivation for its structure, and invite the mathematical community to verify, refine, or refute it. *** UPDATE: Please note two followups have been released https://doi.org/10.5281/zenodo.17345975 and https://doi.org/10.5281/zenodo.17345993 The conjecture was partial flawed however we were able to build further upon AM-GM