  We give bounds on the difference between the weighted arithmetic mean and the
weighted geometric mean. These imply refined Young inequalities and the
reverses of the Young inequality.
  We also study some properties on the difference between the weighted
arithmetic mean and the weighted geometric mean.
  Applying the newly obtained inequalities, we show some results on the Tsallis
divergence, the R\'{e}nyi divergence, the Jeffreys-Tsallis divergence and the
Jensen-Shannon-Tsallis divergence.
