Jensen gap for a real-valued random variable $X \geq 0$ For a real-valued random variable $X \geq 0$,
We have $1 - \frac{1}{1+E \left[x\right]} \geq E \left[ 1 - \frac{1}{1+x} \right]$ (Jensen's inequality).
We want to get a tight constant gap between $1 - \frac{1}{1+E \left[x\right]}$ and $E \left[ 1 - \frac{1}{1+x} \right]$, i.e., $ \lvert  1 - \frac{1}{1+E \left[x\right]} - E \left[ 1 - \frac{1}{1+x} \right] \rvert \leq \epsilon_0$.
Any hints for this inequality?

Taylor expanding $\frac{1}{1+x}$ around $x = E[x]$ gives us
\begin{align*}
\text{Gap} &= \left|1 - \frac{1}{1+E[x]} - E\left[1 - \frac{1}{1+x}\right] \right| \\&= \left|E\left[\frac{1}{1+x}\right] - \frac{1}{1+E[x]} \right| \\
&= \left|\frac{1}{1+E[x]} - E\left[\frac{x - E[x]}{(1 + E[x])^2}\right] +  E\left[\frac{(x - E[x])^2}{(1 + E[x])^3}\right] - \cdots - \frac{1}{1+E[x]}\right| \\
&\le \frac{\text{Var}(x)}{(1 + E[x])^3}
\end{align*}
