# how do I prove this inequallity?

$$m,n$$ are integers bigger than 0.

I need to prove:
$$\frac{1}{\sqrt[n]{1+m}}+\frac{1}{\sqrt[m]{1+n}} \geq 1$$

I tried to multiply both sides by the common factor and raise both sides to the power of $$m*n$$ but it did not work for me, and I have no other idea how to proceed.
I don't want the full solution, but just a hint.
Thank you.

• Muck around with AM. GM??? Or raise the left hand side to a power and get something larger than itself or larger than $1$? – fleablood Oct 26 '18 at 17:16

## 1 Answer

HINT

By Bernoulli's inequality we have

• $$(1+m)^\frac1n \le 1+\frac m n$$

• $$(1+n)^\frac1m \le 1+\frac n m$$

• where is that if and only if statement come from? why is it true? – Omer Oct 26 '18 at 16:05
• Refer to LINK – gimusi Oct 26 '18 at 16:08
• thanks. but i never heard about this inequality, this is the first homework I got in calculus 1 course and this should be pretty basic, is there a simpler way to solve this? – Omer Oct 26 '18 at 16:11
• maybe raise to the power of m*n will work? i tried do this and use the binomial formula, maybe there is something that i am missing.. – Omer Oct 26 '18 at 16:15
• @Omer Notably take a look to the case $(1+x)^r$with$0\le r\le 1$ and $x\ge -1$. Do not hesitate to ask for any other clarification! – gimusi Oct 27 '18 at 8:34