Prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4 \sqrt{2014}$ How to prove that $\sqrt{2011} + \sqrt{2013} + \sqrt{2015} + \sqrt{2017} < 4\sqrt{2014}$ without using calculator?

The function $f(x)=\sqrt{x}$ is strictly concave for $x>0$ so by Jensen's inequality, we have
$$
\frac{1}{4}f(2011)+\frac{1}{4}f(2013)+\frac{1}{4}f(2015)+\frac{1}{4}f(2017)<f\left(\frac{1}{4}(2011+2013+2015+2017)\right)
$$
which is $\sqrt{2014}$.
