Probability of a deviation when Jensen’s inequality is almost tight This is a cross-post to a yet unanswered question in Math StackExchange
https://math.stackexchange.com/questions/3906767/probability-of-a-deviation-when-jensen-s-inequality-is-almost-tight
Let $X>0$ be a random variable. Suppose that we knew that for some $\epsilon \geq 0$,
\begin{eqnarray}
\log(E[X]) \leq E[\log(X)] + \epsilon 
\tag{1} \label{eq:primary}
\end{eqnarray}
The question is: if $\epsilon$ is small, can we find a good bound for
\begin{eqnarray*}
P\left( \log(X) > E[\log(X)] + \eta \right)
\end{eqnarray*}
for a given $\eta > 0$. One bound can be obtained this way:
\begin{eqnarray*}
P\left( \log(X) > E[\log(X)] + \eta \right) &=&  P\left( X > \exp(E[\log(X)] + \eta) \right) \\
& \leq & E[X] / \exp(E[\log(X)] + \eta)  \\
& = & \exp( \log E[X] - E[\log(X)] - \eta ) \\
& \leq & \exp(\epsilon - \eta)
\end{eqnarray*}
where the first inequality follows from Markov’s inequality. This seems like a good bound due to the exponential decay with $\eta$, but upon closer examination it appears that it can be significantly improved. If we have $\epsilon = 0$, then this bounds gives
\begin{eqnarray}
P\left( \log(X) > E[\log(X)] + \eta \right) & \leq & \exp(-\eta)
\tag{2} \label{eq:good_but_not_best}
\end{eqnarray}
However, from Jensen's inequality applied to (\ref{eq:primary}) with $\epsilon = 0$ we obtain $\log(E[X]) = E[\log(X)]$ and therefore $X$ is a constant almost everywhere. As a consequence, for any $\eta>0$,
\begin{eqnarray*}
P\left( \log(X) > E[\log(X)] + \eta \right) = 0.
\end{eqnarray*}
which is (of course) infinitely better than (\ref{eq:good_but_not_best}).
It would appear that a better bound should decay to zero as $\epsilon$ decays, and ideally preserve the exponential decay with $\eta$. Any suggestions?
(I am aware a version of this question has been asked previously Quantitative Version of Jensen's Inequality?)
 A: $\newcommand\ep\epsilon $Let $u:=\eta>0$, so that the probability in question is $P(\ln X>E\ln X+u)$. Note that this probability will not change if we replace there $X$ by $tX$ for any real $t>0$. So, without loss of generality
\begin{equation*}
E\ln X=0,   \tag{-1}
\end{equation*}
and hence your condition (1) can be rewritten as
\begin{equation*}
    EX\le e^\ep, \tag{0}
\end{equation*}
and then the probability in question simplifies to
\begin{equation*}
    P(X>v),
\end{equation*}
where
\begin{equation*}
    v:=e^u>1. 
\end{equation*}
Take now any $z\in(0,v)$ and for all real $x>0$ let
\begin{equation*}
    g(x):=ax-b\ln x+c,
\end{equation*}
where
\begin{equation*}
    a:=a(z):=\frac{1/v}{h(r)},\quad b:=b(z):=az,\quad c:=c(z):=az\ln\frac ze, 
\end{equation*}
\begin{equation*}
    h(r):=1-r+r\ln r,\quad r:=z/v\in(0,1). 
\end{equation*}
Note that the function $h$ is decreasing on $(0,1)$, with $h(1-)=0$. So, $h>0$ on $(0,1)$ and hence $a>0$ and $b>0$. So, the function $g$ is convex on $(0,\infty)$. Moreover,
\begin{equation*}
    g(z)=g'(z)=0, \quad g(v)=1. 
\end{equation*}
It follows that $g(x)\ge1(x>v)$ for all real $x>0$ and hence, in view of (-1) and (0),
\begin{equation*}
    P(X>v)\le Eg(X)=a\,EX+c\le ae^\ep+c. \tag{1}
\end{equation*}
The latter expression, $ae^\ep+c$, in (1) can now be minimized in $z\in(0,v)$, with the minimizer expressed in terms of Lambert's $W$ function.
The suboptimal but simple choice $z=1$ in (1) yields
\begin{equation*}
    P(\ln X>E\ln X+u)=P(X>v)\le\frac{e^\ep-1}{v-1-\ln v}
\end{equation*}
and hence
\begin{equation*}
    P(\ln X>E\ln X+u)\le B_\ep(u):=\min\Big(1,\frac{e^\ep-1}{e^u-1-u}\Big).
\end{equation*}
The simple upper bound $B_\ep(u)$ has both of the desired properties:
(i) for each real $u>0$
\begin{equation*}
    B_\ep(u)\underset{\ep\downarrow0}\longrightarrow0;
\end{equation*}
(ii) uniformly over all $\ep\in(0,1)$ (say)
\begin{equation*}
    B_\ep(u)=O(e^{-u})
\end{equation*}
as $u\to\infty$.
