Problem
When studying Chernoff bound, one result is used without proof and reference, which is $$ (1+x)\log(1+x)-x\geq \frac{x^2}{\frac{2}{3}x+2} $$ I am wondering how this is proved.
What I Have Done
I checked the minimum of LHS and maximum of RHS, this indeed holds. But when it comes to prove this, this sort of check is far from enough.
Something I think relatable is doing some Taylor expansion of LHS, but I did not get the result.
Could someone help me, thank you in advance.
Edit
Take the second-order derivative of $f(x)=(1+x)\log(1+x)-x-\frac{x^2}{\frac{2}{3}x+2}$ gives us $f''(x)=\frac{x^2\, \left(x + 9\right)}{\left(x + 1\right)\, {\left(x + 3\right)}^3}$, which shows the correctness of the answer below.