On a Theorem of V. М. Zolotarev

Let $\xi=\eta=\zeta$, where $\eta$ and $\zeta$ are independent random variables, $\eta$ has the probability density (7) and ${\mathbf E}\exp({\zeta/2})=K<\infty$. It is shown that formula (10) is true if $m\geqq 1$, or if $0<m<1$ and condition (11) which is implied by (12) is satisfied. If ${\mathbf P}\{{\zeta<0}\}=0$, inequality (13) holds for $m\geqq 1$. Formula (14) is true if conditions (15) and (in the case $r>m-1$) (16) are satisfied. An application to the random variable (1), a weighted sum of independent $\chi^2$ random variables, implies a result of V. M. Zolotarev [1].