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?