The unimprovability of moment estimates

Let $\eta$ be a nonnegative random variable. A. M. Zubkov in [Obozrenie Prikl. Prom. Mat., 1 (1994), pp. 638–666 (in Russian)] obtained upper and low estimates for $P\{\eta>0\}$ in the form of a ratio of determinants formed by moments of $\eta$. The low estimates are always nonnegative and the upper estimates can take values from ${[1,\infty)}$. We show that the low and the upper estimates are unimprovable; i.e., for any random variable $\eta\ge 0$ there exist random variables $\zeta\geq 0$ and $\xi\geq 0$ with the same first moments as $\eta$ have, for which $P\{\zeta>0\}$ coincides with the low estimate and $P\{\xi>0\}$ coincides with the minimum of the upper estimate and 1.