Show that the inequality is true | Geometric Mean

Let http://www.physicsforums.com/latex_i.../2932318-0.png be strictly positive numbers. Show that the inequality

http://www.physicsforums.com/latex_i.../2932318-1.png

is true. Where http://www.physicsforums.com/latex_i.../2932318-2.png and http://www.physicsforums.com/latex_i.../2932318-3.png.

The prof hinted that we should use log(1+e^x) and associate r with e^x.

Any insight? I'm not looking for a full solution, just some guidance as to how to approach this problem.