  In this paper, mainly using the convexity of the function
$\frac{a^x-b^x}{c^x-d^x}$ and convexity or concavity of the function
$\ln\frac{a^x-b^x}{c^x-d^x}$ on the real line, where $a>b\geq c>d>0$ are fixed
real numbers, we obtain some important relations between various important
means of these numbers. Also, we apply the obtained results to Ky Fan type
inequalities and get some new refinements.
