The general Jensen's inequality states: $\varphi\left(\mathbb{E}[X]\right) \leq \mathbb{E}\left[\varphi(X)\right]$. I'm wondering if there is a constant $c$ (function of $\varphi$), such that $c\varphi\left(\mathbb{E}[X]\right) \geq \mathbb{E}\left[\varphi(X)\right]$?
More specifically I wan't to show $\log\mathbb{E}[e^X]\leq(e-1)\mathbb{E}[X]$. Here $(e-1)$ would be the $c$ mentioned above.
I've noticed that many other common inequalities have such 'reversing' constants (or at least a corresponding lower bound). Like $\log(1+x)\leq x$, and $\frac{x}{x+1}\leq \log(1+x)$.
Answer
For the particular problem about the exponential function, let $X=-100$, $0$, or $100$ each with probability $\frac{1}{3}$. Then $E(X)=0$, but $E(e^X)$ is large.
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