Let f(x, y) be a positive symmetric n-order homogenous function
defined on R₊xR₊ which is three-time differentiable. Then √(Hf(p,q)Hf(-p,q)) is strictly increasing (decreasing) in p on (0,∞) if J = (x − y)(xI)x < (>)0, where I = (ln f)xy. As applications, H. Alzer’s inequalities are generalized and refined, some new inequalities for logarithmic mean, arithmetic mean and
exponential mean are presented.