# Simple inequality?

## Main Question or Discussion Point

Is there a way to do this without differentiation?

$\left(a+b\right)^{p}$ $\leq a^{p}+b^{p}$

0<p<1 and a,b$\geq$ 0

pulling the a out of the the first part and dividing by it to get

$\left(1+\frac{b}{a}\right)^{p}$$\leq 1+\frac{b}{a}^{p}$

This seems like the way to go but am stuck. Any suggestions? Thanks.

## Answers and Replies

Is there a way to do this without differentiation?

$\left(a+b\right)^{p}$ $\leq a^{p}+b^{p}$

0<p<1 and a,b$\geq$ 0

pulling the a out of the the first part and dividing by it to get

$\left(1+\frac{b}{a}\right)^{p}$$\leq 1+\frac{b}{a}^{p}$

This seems like the way to go but am stuck. Any suggestions? Thanks.
Use the binomial theorem?

p is not an integer though. Not sure binomial thm would work.

0 < p < 1

p is not an integer though. Not sure binomial thm would work.

0 < p < 1
It works.

Did not know that, will do some research.

Did not know that, will do some research.
It is Newton's generalisation that works. It is there on wikipedia.