N > 1 is a natural number.
There are two vectors of N dimension, A and B:
$A = (a_1, a_2, ..., a_N)$ where $a_1>a_2>...>a_N>0$
$B = (b_1, b_2, ..., b_N)$ where $1>b_1>b_2>...>b_N>0$
Let $\lVert A \rVert_p$ denote a $p$-norm of vector A, $\lVert A \rVert_p\equiv\left(\sum_{k=1}^N a_k^p\right)^{1/p}$.
Then, can we prove the following when $p>q$?
$$\frac{\lVert A* B \rVert_p}{\lVert A \rVert_p}>\frac{\lVert A* B \rVert_q}{\lVert A \rVert_q},$$ where $A*B=(a_1b_1,...,a_Nb_N)$. My guess is that since when $p$ is larger the $p$-norm puts more weights on relatively lager elements, a change that bring smaller decreases in these elements will result in a smaller decreases in $p$-norm after the change. Can we prove this mathematically?