Upper bound for $\sum_{i=1}^{N}{x_i^{\beta_i}}$

Question

$\{x_i\}_{i=1}^{N}$ 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.

Answer 1

Let $I$ the indices where $x_i < 1$ and $J$ the indices where $x_i \geq 1$

Let $\beta_{{\text{min}}} = \underset{1 \leq i \leq N}\min{\beta_i} > 1$

$\beta_{{\text{max}}} = \underset{1 \leq i \leq N}\max{\beta_i} > 1$

Then $\sum_{i=1}^{N}{x_i^{\beta_i}} = \sum_{i \in I}^{N}{x_i^{\beta_{i}}} + \sum_{i \in J}^{N}{x_i^{\beta_{i}}} \leq \sum_{i \in I}^{N}{x_i^{\beta_{\text{min}}}} + \sum_{i \in J}^{N}{x_i^{\beta_{\text{max}}}} $

Then using the bound you provided:
$\sum_{i=1}^{N}{x_i^{\beta_i}} \leq \left(\sum_{i \in I}^{N}{x_i}\right)^{\beta_{\text{min}}}+\left(\sum_{i \in J}^{N}{x_i}\right)^{\beta_{\text{max}}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta_{\text{min}}}+\left(\sum_{i =1}^{N}{x_i}\right)^{\beta_{\text{max}}}$

Hence $\sum_{i=1}^{N}{x_i^{\beta_i}} \leq f\left(\sum_{i=1}^{N}{x_i}\right)$

where $f(x)=x^{\beta_{{\text{min}}}}+x^{\beta_{{\text{max}}}}$