2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140364In 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.Selected paper in 14th Seminar on Mathematical Analysis and its Applications, February 4-5, 2004, Iran University of Science and Technology, Tehran, IranClassical Analysis and ODEs26D15, 26D15, 26A06, 39B62text