  In the present note we give an elementary proof of the necessary and
sufficient condition for a univariate function to belong the class $\mathcal
N_\varkappa^+$. This class was introduced mainly to deal with the indefinite
version of the Stieltjes moment problem (and corresponding $\pi$-Hermitian
operators), although it is applicable beyond the original scope. The proof
relies on asymptotic analysis of the corresponding Hermitian forms. Our result
closes a gap in the criterion given by Krein and Langer in their joint paper of
1977. The correct condition was stated by Langer and Winkler in 1998, although
they provided no proper reasoning.
