$m,n$ 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.

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


By Bernoulli's inequality we have

  • $(1+m)^\frac1n \le 1+\frac m n$

  • $(1+n)^\frac1m \le 1+\frac n m $

  • $\begingroup$ where is that if and only if statement come from? why is it true? $\endgroup$ – Omer Oct 26 '18 at 16:05
  • $\begingroup$ Refer to LINK $\endgroup$ – gimusi Oct 26 '18 at 16:08
  • $\begingroup$ 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? $\endgroup$ – Omer Oct 26 '18 at 16:11
  • $\begingroup$ 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.. $\endgroup$ – Omer Oct 26 '18 at 16:15
  • 1
    $\begingroup$ @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! $\endgroup$ – gimusi Oct 27 '18 at 8:34

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