Let $Z$ be a random variable with finite moment-generating function $M\_Z(\theta):=E[e^{\frac{1}{\theta}Z}]<\infty$ for all $\theta > 0$, and for $\delta \in (0,1]$, define
$C\_Z^\delta := \inf\_{\theta>0}\theta\log M\_Z(\theta) - \theta\log(\delta)$,
and $SVP\_Z^\delta:=E[Z] + \sqrt{(1/\delta)Var[Z]}$. By the Chernoff inequality, one has $P(Z \ge C^\delta\_Z) \le \delta$. Also, one notes that if $Z$ is Gaussian, then $C^\delta\_Z = SVP\_Z^\delta$.


Question
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Are there any interesting bounds on the difference $C^\delta\_Z - SVP\_Z^\delta$ ? You may assume $Z$ is bounded almost surely, say $P(|Z| \le R)= 1$.



Bounds on difference between "logsumexp" and variance?
By [Theorems 3.3 and 3.4](https://www.mdpi.com/2227-9091/2/3/349), 
$$
C\_Z^\delta\ge C\_{Z;\alpha}^\delta:=\inf\_{\theta\in\mathbb R}\Big(\theta+\frac{\|(Z-\theta)\_+\|\_\alpha}{\delta^{1/\alpha}}\Big)
$$
for any $\alpha\in(0,\infty)$, where $\|X\|\_\alpha:=(E|X|^\alpha)^{1/\alpha}$. This lower bound on $C\_Z^\delta$ may be especially useful when $M\_Z(\theta)=Ee^{Z/\theta}=\infty$ for all $\theta>0$, while $EZ\_+^\alpha<\infty$ for some $\alpha\in(0,\infty)$, because for all $\alpha\in(0,\infty)$ we still have 
$$P(Z\ge C\_{Z;\alpha}^\delta)\le\delta,
$$
again by the monotonicity-in-$\alpha$ part of mentioned Theorem 3.4 and formula (3.3) in that paper. 


