In Theorem 14.2 of his book Stochastic Limit Theory Davidson establishes the bound
$$E[\big|E[X|\mathcal F]-E[X]\big|]\leq 4M\alpha$$
where $\alpha,M\geq0$ are constants, and $|X|\leq M$. He then writes:
"Since $\big|E[X|\mathcal F]-E[X]\big| \leq 2M$ it follows that, for $p\geq1$, $\|E[X|\mathcal F]-E[X]\|_p \leq 2M(2\alpha)^{1/p}$"
Why is this true?
Let $Y:=\left\lvert\mathbb E\left[X\mid\mathcal F\right]-\mathbb E\left[X \right]\right\rvert$. It is already shown that $\mathbb EY\leqslant 4M\alpha$. Moreover, we know that $0\leqslant Y\leqslant 2M$ hence $$ \mathbb E\left[Y^p\right]=\mathbb E\left[Y\cdot Y^{p-1}\right]\leqslant \mathbb E[Y] (2M)^{p-1}\leqslant 4M\alpha(2M)^{p-1} =2^{p+1}M^p\alpha. $$ Then take the power $1/p$ to get the wanted inequality.