Let $w$ be an $n$-dimensional real vector with non-negative entries, and let $t>1$. I want to bound $$\frac{\sum_{i=1}^n w_i \times \sum_{i=1}^n w_i^{2t-1}}{(\sum_{i=1}^n w_i^t)^2}$$ In the language of $l_p$ norms of vectors, this can be rewritten as $$\frac{||w||_1 \times ||w||_{2t-1}^{2t-1}}{(||w||_t^t)^2}$$
When the entries of the vector are non-negative but otherwise completely arbitrary, this ratio can be as bad as $n^{1-1/t}$, asymptotically. This can be proved using Holder's inequality. For example, when $(n-1)$ entries have value $n^{1-1/t}$ but the remaining one has value $n$.
However, suppose we bound how large each entry of the vector can be, say $w_i\leq l$ for all $i$. What kind of bound would we get for the above ratio? How does one prove such results?
Thanks.