$\{x_i\}_{i=1}^{N}$ a sequence of positive real numbers and $\{\beta_i\}_{i=1}^{N} $ are real numbers such that $\underset{1 \leq i \leq N}\min{\beta_i}$ > 1.
Is it possible to find an upper bound of the form:
$$ \sum_{i=1}^{N}{x_i^{\beta_i}} \leq f\left(\sum_{i=1}^{N}{x_i}\right) $$
where $f$ is a smooth function.
I did try to bound $\sum_{i=1}^{N}{x_i^{\beta_i}} $ by $\sum_{i=1}^{N}{x_i^{\alpha}} $ and then using the answer in this post, one can find an answer to the question. However, the parameter $\alpha$ that I found depends on the fact whether $x_i < 1$ or $x_i >1$. In fact,
If $x_i < 1$, $\alpha = \underset{1 \leq i \leq N}\min{\beta_i}$,
If $x_i > 1$, $\alpha = \underset{1 \leq i \leq N}\max{\beta_i}$.
Is it possible to find an $\alpha$ independent of $i$ or if someone has another idea on how to find the function $f$ if possible.
Thanks.