Let $R$ be a positive integer, $\mathcal{X}$ be the sample space and $x \in \mathcal{X}$ be an event of the sample space; $P(x)$ denotes the probability of occurrence of event $x$. The problem is to upper bound the following expression:
$$ \sum_{x\in \mathcal{X}}x(P(x))^{\frac{1}{R}}, $$ in terms of $$\sum_{x\in \mathcal{X}}xP(x).$$
Is it possible to get a tighter bound for the expression?