Inequality :$\left(\frac{x}{x+1}\right)^{\tan^2(x)}+\left(\frac{1}{1+3}\right)^{\tan^2(1)}+\left(\frac{3}{x+3}\right)^{\tan^2(3)}>\frac{40}{39}$

Question

Claim :

Let $\frac{13}{10}\leq x<\frac{\pi}{2}$

$$f(x)=\left(\frac{x}{x+1}\right)^{\tan^2(x)}+\left(\frac{1}{1+3}\right)^{\tan^2(1)}+\left(\frac{3}{x+3}\right)^{\tan^2(3)}>\frac{40}{39}$$

The function in the RHS is close to a line and the choice of $\frac{40}{39}$ is because it's a simple constant and rational .As you can see I'm inspired by the Nesbitt's inequality so it's not totally random .

To prove I have tried convexity with the second derivative of $f(x)$ wich is positive but I cannot prove it currently .

Any idea to prove it ?

Thanks !

Answer 1

Hint:

Fact 1: If $0 < x < \frac{\pi}{2}$, then $g(x) = \tan^2 x \ln (\frac{x}{x+1})$ is strictly decreasing.
(Hint: Take derivative and use $\tan x \ge x$.)

Now, by Fact 1, we have $f(x) \ge f(\pi/2) > \frac{40}{39}$. We are done.