# Connection between Bernoulli's inequality and Young's inequality .

Working on a question Proof that if $x,y>0$ and $x+y=1$, then $(2x)^{\frac 1 x}+(2y)^{\frac 1 y}\leq 2$ . I find a connection between Bernoulli's inequality and Young's inequality :

Let $$0< r\leq 1$$ and $$x\geq0$$ then prove that : $$(1+x)^r\leq 1+rx$$

$$1(1+x)^r\leq \frac{(1+x)^{rp}}{p}+\frac{1}{q}$$

With $$\frac{1}{q}+\frac{1}{p}=1$$ with $$p,q>0$$

we put now $$p=\frac{1}{r}$$ to get :

$$(1+x)^r\leq (1+x)r+1-r$$

Wich is the desired inequality .

Question

Is it fine ? Can we improve the situation ?