Preliminaries: let $f_n(x)\,, n=0,\ldots,\infty$ denote the Hermite functions, which are of the form $$ f_n (x)=\#\, e^{-\frac{x^2}{2}} H_n(x) \,,$$ where $H_n(x)$ are the physicists' Hermite polynomials, $\#$ denotes a normalisation constant such that $$ \int_\mathbb{R} | f_n |^2 dx = 1 \,.$$
Let $H[g]$ denote the differential entropy of $g$, defined as $$ H[g] = -\int_\mathbb{R} |g(x)|^2 \log |g(x)|^2 dx \,,$$ where again $\int |g|^2 =1$.
Question 1: Can it be proven that among all Hermite functions the one with the least entropy is the one with $n=0$? For the first few it is easy to check that this holds. There exist some asymptotic results for the entropy of the Hermite polynomials and the behaviour suggests that the entropy is increasing with $n$. However, I am not aware of a complete proof. I do not need the value of the entropy (probably hard to get an analytic expression in general) but just some argument that it must increase, or even that it just does not decrease.
Question 2: Let's assume now that one can show Q1. By counter example it follows that $$H[a f_0(x) + b f_j (x)] \ngeq H[f_0(x)]\,, $$ where $a,b\in \mathbb{C}$ such that $|a|^2 +|b|^2 =1$. Specifically, for $j=1,2$ and values of $a\sim 0.8$, one can see that the last inequality is violated. However, I have numerical evidence that the following inequality holds: $$H[a f_0(x) + b f_j (x)] + H[a f_0(x) + r^j b f_j (x)] \geq 2H[f_0(x)]\,, $$ where $r$ is a fourth root of unity, i.e. $r=e^{i\pi/2}$. Is there an elementary way to establish such an inequality based on the form of the two functions $a f_0(x) + b f_j (x)$ and $a f_0(x) + r^j b f_j (x)$ and maybe properties of the entropy?