I found this inequality beautiful let me share it :
Let :
$$d=\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\cdot\cdot}}}},a=\frac{\sqrt{5}-1}{2},b=\frac{\sqrt{5}+1}{2}$$
Then :
$$\left(\frac{1}{1+\frac{a}{1+\frac{2a^{2}}{1+\frac{3a^{3}}{1+\cdot\cdot\cdot}}}}+\frac{1}{1+\frac{b}{1+\frac{2b^{2}}{1+\frac{3b^{3}}{1+\cdot\cdot\cdot}}}}\right)^2<\pi-\ln\pi$$
I have knowledge of continued fractions so if you could provide me with some hint(s), that would be preferable .
Let $v$ as in my comment below then it seems we have :
$$|r-t|<d^2-\pi+\ln\pi$$
Where $r$ is the expression $v$ with the value $a=\sqrt{5}$ and $t$ the value $a=\sqrt{\pi/4}$.
Remark :
It might be interesting to evaluate the odd and even case I mean the difference so the two differences in $v$ expression perhaps it's equal... to $0$
How do I show this? Do we have a closed form expression for $d$?
Just computing the numbers for $$d=\big(f(a)+f(b)\big)^2$$ Using $1000$ levels, the numerical values are $$f(a)=0.70971940484203666255310483661179592566676895106748$$ $$f(b)=0.69762814978893353418889274755209993243313281242287$$ $$d=1.9806271395257716424616215274705710807886558152997$$
which is much smaller than $\pi-\log(\pi)$ (the difference is $\sim 0.016235$)
None of these numbers has been recognized by the inverse symbolic calculators I tried.
However, searching for a simple approximation $$\color{blue}{\large d \sim \pi-\log(\pi)-\frac 1{10 e}\Omega ^{\sqrt[3]{3}}}$$ which is $\color{red}{1.9806271}408$