Prove that $(\frac{\pi}{3})^{\frac{3}{\pi}}+(\frac{3}{\pi})<2$

Question

Prove that :

$$\left(\frac{\pi}{3}\right)^{\frac{3}{\pi}}+\frac{3}{\pi}<2$$

Straightforward proof :

Since the function $f(x)=(x)^{\frac{1}{x}}+\frac{1}{x}$ is decreasing on$\left[1,\frac{\pi}{3}\right]$

We get :

$$f\left(\frac{\pi}{3}\right)=\left(\frac{\pi}{3}\right)^{\frac{3}{\pi}}+\frac{3}{\pi}<f(1)=2$$

So we get an almost integer easily .

My questions :

Have you an alternative proof ?

Can we find other almost integer with this way?

Thanks a lot for all your contributions .

Answer 1

Bernoulli's inequality for exponents $\in[0,1]$ $$\left(\frac{\pi}{3}\right)^{\frac{3}{\pi}}+\frac{3}{\pi}= \left(1+\frac{\pi-3}{3}\right)^{\frac{3}{\pi}}+\frac{3}{\pi} \leq 1+\frac{\pi-3}{3}\cdot \frac{3}{\pi}+\frac{3}{\pi}=\\ 1+1-\frac{3}{\pi}+\frac{3}{\pi}=2$$