I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the condition and my problem. Suppose $Q$ is a probability measure and $r>1$, $f,g,F$ are measurable functions and $Qf=\int f dQ$. Under the conditions below
$|f|\le F$
$F$ is integrable
$\int|f-g| dQ<2\int F dQ$
the proof of the theorem said $$ Q|f-g|^r\le Q|f-g|(2F)^{r-1}. $$
I don't know how to derive this inequality. I tried to use $C_r$-inequality, $$ \begin{align} Q|f-g|^r&=\int|f-g||f-g|^{r-1}dQ \\ &\le\int|f-g| \cdot 2^{r-2}(|f|^{r-1}+|g|^{r-1})dQ\\ &\le \int|f-g| \cdot 2^{r-2}F^{r-1}dQ+\int|f-g| \cdot 2^{r-2}|g|^{r-1}dQ. \end{align} $$ Hence I need to prove $\int|f-g| \cdot 2^{r-2}|g|^{r-1}dQ\le \int|f-g| \cdot 2^{r-2}F^{r-1}dQ$. From condition 1 and 3, it can be shown $Q|g|\le 3QF$ by the triangle inequality. However I cannot proceed further with the proof. I guess this should be obvious and easy, but I am really confused and have spent the whole day on this point.
Any help would be appreciated.
