Zinn's "doubling" conjecture on weighted sums of independent Rademacher random variables

Question

Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let $S:=a_1\eta_1+\dots+a_n\eta_n$, and let $T$ be an independent copy of $S$. Does then the inequality $$(*)\qquad E f_p(S)\le E f_p\Big(\frac{S+T}{\sqrt2}\Big)$$ hold for all real $p\ge2$, where $f_p(x):=|x|^p$?

This conjecture was communicated to me by Joel Zinn quite some time ago, and I think it deserves to be more broadly known. A motivation behind it was to obtain an alternative (and hopefully easier) proof of Haagerup's inequality $$E f_p(S)\le E f_p(Z)$$ Haagerup, which indeed easily follows from $(*)$ by the central limit theorem; here $Z$ is a standard normal r.v. For $p=2$, $(*)$ is trivial. For $p\ge3$, $(*)$ is easily proved; in particular, it follows immediately from Corollary 2.5 in T^2.

So, actually the question is only about $p\in(2,3)$. In that case, in view of Lemma 2.2 in Figiel et al, it would be enough to prove $(*)$ with $g_t$ in place of $f_p$ (for all real $t>0$), where $g_t(x):=|x|^3-\max(0,|x|-t)^3$.

Answer 1

1. Alas, the approach through the functions $g_t$ cannot possibly work. That is seen if one takes e.g. $n=2$, $a_1=4/5$, and $a_2=t=3/5$.

2. Based on numerical evidence, the following approach promises to work. Without loss of generality (wlog) $n\ge2$ and $a_1\ge\dots\ge a_n\ge0$. Let $a:=a_1$, $b:=a_2$, and $Y:=|S-a_1\eta_1-a_2\eta_2|$. Then $$(1)\quad E Y^2=1-a^2-b^2$$ and $$(2)\quad EY^4=3\Big(\sum_3^n a_i^2\Big)^2-2\sum_3^n a_i^4\ge3(1-a^2-b^2)^2-2(1-a^2-b^2)\times\min(b^2,1-a^2-b^2), $$ since $\max_3^n a_i^2\le\min(b^2,1-a^2-b^2)$. By induction, it is enough to show that $$ (3)\quad E|S_2+Y|^p\le E\Big|\frac{S_2+T_2}{\sqrt2}+Y\Big|^p$$ for all $p\in(2,3)$ and nonnegative r.v.'s $Y$ subject to conditions (1) and (2), where $S_2:= \eta_1 a+\eta_2 b$ and $T_2$ is an independent copy of $S_2$. At that, by well-known results (see e.g. Hoeffding55), wlog the r.v. $Y$ takes at most $3$ values (say $0\le u\le v\le w$ with probabilities $r,s,1-r-s$).

Thus, the problem is reduced to a calculus problem on proving (say) the nonnegativity of a function of $8$ variables $p,a,b,u,v,w,r,s$ subject to a finite number of restrictions on the values of these variables. So, in principle this problem is solvable, but seems very involved computationally.

Addendum: Unfortunately, other numerical evidence shows that conditions (1) and (2) on $Y$ are not enough for (3) to hold in general. For instance, if $a = b = 11/21$, $p = 93/46$, $u = 0$, $v = 11/95$, $w = 71/61$, and $r\approx0.642$ and $s\approx0.025$ are such that $E Y^2=1-a^2-b^2$ and $EY^4=3(1-a^2-b^2)^2$, then the difference between the right-hand and left-hand sides of (3) is $\approx-0.000163$.