Let a pair of random variables $(X, Y)$ over a finite product space $\mathcal{X}\times \mathcal{Y}$ be given. The conditional expectation operator is defined as 
$$(T\_Yf) (y):=\mathbb{E}[f(X)|Y=y],$$
where $f$ is a real-valued function acting on $\mathcal{X}$.


It is well known the the operator $T$ is contractive in $L^p$-norm for $p\geq 1$, i.e., 
$$||Tf||\_p\leq ||f||\_p,$$
which can be easily proved using convexity of the map $t\mapsto t^p$ and Jensen's inequality.


I am looking at the map $p\mapsto \frac{||Tf||\_p}{||f||\_p}$ for $p\geq 1$.
I am trying to see if this map is monotonic or not. Any idea? 



Monotonicity of a ratio of conditional expectation operator
What if $X=\{1,2\}$ and $Y=\{1,2,3\}$. Equip both $X$ and $Y$ with the uniform probability measure. Set $f(x,1)=1$, $f(x,2)=x$ and $f(x,3)=2$. 
Then $Tf(x,1)=1$; $Tf(x,2)=1.5$ and $Tf(x,3)=2$. 


Notice that $\|f\|\_1=\|Tf\|\_1=1.5$ and $\|f\|\_\infty=\|Tf\|\_\infty=2$,
but $\|f\|\_2>\|Tf\|\_2$. 


