Let $a,b \ge 0$ and $\frac{m}{n} \le 1$ then $a^{\frac{m}{n}} + b^{\frac{m}{n}} \ge (a+b)^{\frac{m}{n}}$.
How to prove this inequality.
If $\frac{m}{n} = \frac{1}{2}$, I know how to prove this.
The function $f(x) = x^\frac{m}{n}, x \ge 0$ is increasing for $\frac{m}{n} \le 1$.
How can I use the above fact?
Can you help me out.
This question : For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
has an answer for $a+b = 1$.