I can imagine that there is a confusion in the definition of $\beta_r(X)$. For your formula it should be defined as $E(|X|^r)^{1/r}$ and not as $E(|X|^r)$. Can you find a counterexample in the homogeneity if $E$ is a Dirac mass in $0$ and $F$ a scaled random variable?

Nicolas

There is a confusion in the definition of $\beta_r(X)$. It does not match with the estimate. The estimate of the Encyclopedia should be written for $\gamma_r(X)=E(|X|^r)^{1/r}$ in place of $\beta_r(X)=E(|X|^r)$. The inequality that you can prove is $$d_L(E,F)\leq\gamma_r(F)^{\frac{r}{r+1}}.$$

On the other side, as we will see the same inequality is wrong for $\beta_r$. Let $F_\epsilon$ be uniform on $\{-\epsilon,+\epsilon\}$ and set $E$ the Dirac mass in $0$. You get $d_L(E,F_\epsilon)=\epsilon$ for every $\epsilon<1/2$ but also $\beta_r(F_\epsilon)=\epsilon^r$. Then $d_L\leq \beta_r^{r/(r+1)}$ simply writes $$\epsilon\leq \epsilon^{\frac{r^2}{r+1}}.$$ This is of course wrong, for instance for $r=2$.

Conclusion: use $\gamma_r$ as definition of the $r$-th moment, or use $\beta_r$ and replace the estimate of the Encyclopiedia by your estimate.

Nicolas