  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.
