Paltanea type theorems on estimation by positive discrete functionals

The article is concerned with inequalities of the type
\begin{equation*} |F(f)-F(e_0)f(x)| \le F(e_0)\omega_2(f, h), \end{equation*}
there $F$ is a functional of the form $F(f)=\sum\limits_{y \in Y}\gamma(y)f(y)$, and $Y$ is an at most countable set with no accumulation points on $\mathbb{R}$, $\gamma : Y \to (0, \infty)$.