ratio of consecutive $l^p$ norms of cumulative distribution functions

Question

Suppose $G$ is a continuously differentiable CDF over $[a,b]$.

I am trying to figure out whether the following inequality is true for any $n\in\mathbb{N}$:

$$\begin{equation} n\int_{a}^{b}G(x)^ndx\leq (n+1)\int_{a}^{b}G(x)^{n+1}dx \end{equation}$$ I can re-write the above condition as: \begin{equation} \frac{n}{n+1}\leq \frac{\|G\|_{n+1}^{n+1}}{\|G\|_{n}^{n}} \end{equation} Using Holder's inequality we can find lower bound on the right hand side but that doesn't help much. I am not sure whether this is true or not. I did do some computations and it holds for almost any distribution I can think of. Any advice would be greatly appreciated!

Answer 1

This is not true. Take $[a,b] = [0,1]$, $f(x) = \frac{1}{2}$ and $n=2$. We have $$ \frac{1}{2} = n \int f(x)^n > (n+1)\int f(x)^{n+1} = \frac{3}{8}. $$

Using a smooth CDF $G$ approximating $f$, you can conclude.

Answer 2

Let me reformulate the counterexample sketched by @Olivier: it suffices to take the cdf of a Bernoulli RV with $p=\dfrac12$. Its pdf is $\frac12(\delta_0+\delta_1)$ and cdf is defined by $F(x)=1/2$ on $(0,1)$.

Then, $N_n=n \int_0^{1}dx/2^n=\dfrac{n}{2^n}$ constitutes a decreasing sequence for $n>1$.