Consider the sets of i.i.d. random variables $\{ X_i \}_{i = 1}^n$ and $\{ Y_i \}_{i = 1}^n$, which are independent of each other. Denote their sample means as $\overline{X} = \frac{1}{n} \sum_{i = 1}^n X_i$ and $\overline{Y} = \frac{1}{n} \sum_{i = 1}^n Y_i$, and their population means as $\mu_X$ and $\mu_Y$, respectively. We seek appropriate upper bounds $U_X$ and $U_Y$ for the population means such that $$ \mathrm{P} (\mu_X - \overline{X} > U_X) \leq \alpha / 2, \qquad \mathrm{P} (\mu_Y - \overline{Y} > U_Y) \leq \alpha / 2. $$ where $U_X$ and $U_Y$ are positive functions incorporating distribution information.
We wish to find a suitable upper bound $U_Z$ for the product of the population means, $\mu_Z = \mu_X \mu_Y$, which represents the expected value of $Z_i = X_i Y_i$. We want to find a function $U_Z$ in terms of $\overline{X}, \overline{Y}, \overline{Z}, U_X, U_Y$ satisfying $$ \mathrm{P} (\mu_Z- \overline{Z} > U_Z) \leq \alpha. $$
PS: Previously, I tried $U_Z = \overline{X} U_Y + \overline{Y} U_X + U_X U_Y$ as $$ \mathrm{P} (\mu_Z - \overline{Z} > {\overline{X} U_Y + \overline{Y} U_X + U_{X} U_Y}) \leq \mathrm{P} (\mu_X \mu_Y - \overline{X} \cdot \overline{Y} > {\overline{X} U_Y + \overline{Y} U_X + U_{X} U_Y}) = \mathrm{P} \big( (\mu_X - \overline{X}) (\mu_Y - \overline{Y}) > U_X U_Y \big) \leq \alpha. $$ But it does not work, as I see we cannot guarantee that $\overline{Z} \geq \overline{X}\cdot \overline{Y}$. But I trust there is a similar expression.