What's the lower bound for this quantity?

Question

Suppose $p$ is a discrete distribution with $n$ values and the random variable $x$ satisfies $\mathbb{E}_p[x] = 0$ and $|x| < \infty$. Given $\alpha \in (0,1)$, does there exist a lower bound for the quantity $$ k:=\frac{\log \left( \mathbb{E} _p\left[ e^{\alpha x} \right] \right)}{\log \left( \mathbb{E} _p\left[ e^x \right] \right)} $$ that is only related with $\alpha$ ? By Jensen inequality, we know that $k \le \alpha$. I wonder what's the lower bound for $k$.

Answer 1

The exact lower bound on $k$ is $0$.

Indeed, let $a:=\alpha$. By Jensen's inequality, $\ln Ee^{ax}\ge0$ for all real $a$. So, $k\ge0$ when $k$ is defined.

On the other hand, let $y$ be any random variable (r.v.) such that $Ee^{ay}<\infty=Ee^y$ for $a\in(0,1)$. One can easily construct a zero-mean r.v. $x$ taking only finitely many values such that $Ee^{ax}$ be however close to $Ee^{ay}<\infty$ and $Ee^x$ be however close to $Ee^y=\infty$, and then the $k$ for $x$ wil be however close to $0$. $\quad\Box$

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More explicitly, suppose that $P(x=v)=\frac u{v+u}=1-P(x=-u)$, where $u$ and $v$ are positive real numbers. Then $Ex=0$ and $$\lim_{v\to\infty}\lim_{u\downarrow0}k =\lim_{v\to\infty}\frac{e^{av}-1-av}{e^v-1-v}=0.$$ So, $k$ can be however close to $0$.