The Weighted Arithmetic Mean-Geometric Mean Inequality is Equivalent to the H\"{o}lder Inequality

In the current note, we investigate the mathematical relations among the weighted arithmetic mean-geometric mean (AM-GM) inequality, the H"{o}lder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM-GM inequality, the weighted power-mean inequality and the H"{o}lder inequality are fully achieved. The new results are more generalized than those of previous studies.