Let $a\in S^d$ and $b\in S^{d-1}$ be points chosen uniformly at random on the $(d+1)$ and $d$-dimensional spheres.


I'm interested in showing some inequalities regarding their norms, the simplest being:



> 
> **How to show that $\mathbb E\left[\frac{||a||\_1}{\sqrt {d+1}}\right] \le \mathbb E\left[\frac{||b||\_1}{\sqrt d}\right]$?**
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> 
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Next, I'm looking for:



> 
> **How to show that $\mathbb E\left[\frac{||a||\_1^2}{{d+1}}\right] \le \mathbb E\left[\frac{||b||\_1^2}{ d}\right]$?**
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> 
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We verified, using a monte carlo simulation, that these hold for all small-ish $d$ values, and that they hold as equalities when $d$ tends to $\infty$.


This question is somewhat related to my [previous question](https://mathoverflow.net/questions/391195/is-mathbb-e-left-fracdx-12-right-o1-for-all-d-in-mathbb-r-w).



Let $a\in S^d$, $b\in S^{d-1}$ be uniform on the spheres. How to show $\mathbb E[\frac{||a||_1}{\sqrt {d+1}}] \le\mathbb E[\frac{||b||_1}{\sqrt d}]$?
$\newcommand{\Ga}{\Gamma}$Your first inequality is true, for each $n:=d\ge2$.
Note that $a$ and $b$ equal, respectively, $X\_{n+1}$ and $X\_n$ in distribution, where
\begin{equation\*}
 X\_n:=G/|G|,
\end{equation\*}
$G=(G\_1,\dots,G\_n)$ is a standard Gaussian random vector in $\mathbb R^n$, and $|G|$ is the Euclidean norm of $G$. So,
\begin{equation\*}
 E\|X\_n\|\_1=n\,EY,
\end{equation\*}
where $Y:=|G\_1|/|G|$, so that $Y^2$ has the beta distribution with parameters $1/2,(n-1)/2$, and hence
\begin{equation\*}
 \frac{E\|X\_n\|\_1}{\sqrt n}=f\_n:=\frac{\sqrt n}{\sqrt\pi}\,\frac{\Ga(n/2)}{\Ga((n+1)/2)}, 
\end{equation\*}
and your first inequality means that
\begin{equation\*}
 r\_n:=f\_n/f\_{n+1}\overset{\text{(?)}}\ge1. \tag{$\*$}
\end{equation\*}


Note that
\begin{equation\*}
 \rho\_n:=\frac{r\_{n+2}}{r\_n}=\frac{(n+2)^{3/2}}{(n+1)^{3/2}}\,\sqrt{\frac{n}{n+3}}<1
\end{equation\*}
for $n>0$, because
\begin{equation}
 \rho\_n^2-1=-\frac{2 n+3}{(n+1)^3 (n+3)}<0.
\end{equation}
So, $r\_{n+2j}$ is decreasing in $j\in\{0,1,\dots\}$, for each $n>0$.
Also, it is easy to see that $r\_n\to1$ (as $n\to\infty$). So, $r\_n>1$ for all $n>0$, that is, ($\*$) holds, as desired.




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Concerning the second inequality, Pierre PC showed in a comment that $E\|X\_n\|\_1^2=1+(n-1)2/\pi$. Hence, the ratio
\begin{equation}
 \frac{E\|X\_n\|\_1^2}{n}=\frac2\pi+\frac{1-2/\pi}n
\end{equation}
is decreasing in $n$, which means the the second inequality holds as well.




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Asking multiple questions in one post is not encouraged on MathOverflow, I think.


