If we have a real random variable $X$ such that $a\leq X\leq b$ almost surely, we can establish the following inequality:
\begin{align}
\mathbb{E}\left[\exp\Big(t(X-\mathbb{E}[X])\Big)\right]\leq\exp\left(\frac{1}{8}t^2(b-a)^2\right).
\end{align}
Is there a similar bound for a complex random variable $Y$ such that it satisfies the condition of $r\_1\leq |Y|\leq r\_2$ with almost certain probability?



Hoeffding's Lemma for bounded complex random variables?
Restricting $Y$ to an annulus doesn't seem useful as any bounds are likely to be satisfied also inside the annulus.


A bound with $Y$ restricted to a disk, or more generally to a region with bounded diameter, appears in Mikhail Isaev and Brendan McKay. "On a bound of Hoeffding in the complex case." Electron. Commun. Probab. 21 1-7, 2016. <https://doi.org/10.1214/16-ECP4372>


Free copy [here](https://arxiv.org/pdf/1603.00613.pdf).


The proof is different from the real case because the exponential function is not concave in the complex plane.


The complex condition most similar to the real condition $a\le X\le b$ is bounded diameter:
$$\mathrm{diam}(Y) = \inf\{c\in\mathbb{R}\,:\,
 P(|Y\_1-Y\_2|>c)=0\},$$
where $Y\_1,Y\_2$ are independent copies of $Y$.
We show that
$$|\mathbb{E}e^{Y-\mathbb{E}Y}-1| \le 
e^{\mathrm{diam}(Y)^2/8}-1,$$
where $\frac 18$ is the best possible constant.
For diameter less than 3 we give the optimum bound.


