Inequality

Let $a, b>0$. Prove that:

$$\left ( 1+ \frac{a}{b} \right )^{2014}+ \left ( 1+ \frac{b}{a} \right )^{2014}\geq 2^{2015}$$

Solution

Using the AM-GM inequality we have successively:

$$\begin{aligned}
\left ( 1+ \frac{a}{b} \right )^{2014}+ \left ( 1+ \frac{b}{a} \right )^{2014} &\geq 2\sqrt{\left [ \left ( 1+ \frac{a}{b} \right )\left ( 1+ \frac{b}{a} \right ) \right ]^{2014}} \\
 &= 2\sqrt{\left ( \frac{a}{b}+ \frac{b}{a}+ 2\right )^{2014}}\\
 &= 2\sqrt{\left [ \left(\sqrt{\frac{a}{b}}+ \sqrt{\frac{b}{a}} \right)^2\right ]^{2014}}\\
 &= 2\left ( \sqrt{\frac{a}{b}}+ \sqrt{\frac{b}{a}} \right )^{2014}\\
 &\geq 2^{2015}
\end{aligned}$$

The exercise can also be found in mathematica.gr