Ky Fan's inequality seems rather beautiful. The most beautiful proof can be found here 

<cite authors="Jovanović, Milan V.; Pogány, Tibor K.; Sándor, József">_Jovanović, Milan V.; Pogány, Tibor K.; Sándor, József_, [**Notes on certain inequalities by Hölder, Lewent and Ky Fan**](http://dx.doi.org/10.7153/jmi-01-07), J. Math. Inequal. 1, No. 1, 53-55 (2007). [ZBL1147.26011](https://zbmath.org/?q=an:1147.26011), [MR2347705](https://mathscinet.ams.org/mathscinet-getitem?mr=2347705).</cite>