Ky Fan's inequality seems rather beautiful. The most beautiful proof can be found here
Jovanović, Milan V.; Pogány, Tibor K.; Sándor, József, Notes on certain inequalities by Hölder, Lewent and Ky Fan, J. Math. Inequal. 1, No. 1, 53-55 (2007). ZBL1147.26011, MR2347705.
