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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 $\frac{x}{x+1}\leq \log(1+x)$.

Thomas Ahle
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    your inequality won't hold without extra assumptions on $X$. take $X$ to be $1$ or $-1$ with probabilities $1/2$ for example – mm-aops Jun 02 '14 at 16:31
  • I was taking $X$ to be positive. I guess this settles whether there is a constant in general. – Thomas Ahle Jun 02 '14 at 16:33
  • Coud I ask have you figured out what kind of inequality we could have for $E[e^X]<?$ – null Apr 11 '20 at 20:26
  • @NathanExplosion Looking at it now, it is obvious that $E[\phi(X)]$ can be arbitrarily large. Even with $\phi(x) = x^2$ it might not exist. But for sub-gaussian $X$ I suppose we get $\log E[e^X] \le E[X] + V[x]/2$. – Thomas Ahle Apr 12 '20 at 08:28
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    @ThomasAhle thanks, so it really depends on the situation. – null Apr 12 '20 at 08:30
  • @ThomasAhle Usually one would consider $E[e^{\lambda X}]$, and then you get different bounds depending on whether that expression is finite for all $\lambda$ or only for $\lambda \leq \lambda_0$. The bound you mention is indeed the definition of a sub-Gaussian random variable. For more on the topic, see for example Chapter 2 of Boucheron, Lugosi, Massart (2013) - Concentration inequalities - a nonasymptotic theory of independence. – BookYourLuck May 17 '20 at 12:04

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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.

André Nicolas
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  • Thank you. Sorry for the bad question. – Thomas Ahle Jun 02 '14 at 16:34
  • There are interesting questions underneath, such as finding natural conditions on the distribution of $X$ that give your desired inequality. – André Nicolas Jun 02 '14 at 16:38
  • It arrised from trying to prove Chernoff bounds for general random variables, instead of just Poisson trials. I'll have to see if I can find a correct general definition. – Thomas Ahle Jun 02 '14 at 16:45