Functions of classes $\mathcal N_\varkappa^+$

In the present note we give an elementary proof of the necessary and sufficient condition for a univariate function to belong the class $\mathcal N_\varkappa^+$. Although this class was introduced mainly to deal with the indefinite version of the Stieltjes moment problem (and corresponding $\pi$-Hermitian operators), it can be useful far beyond the original scope. Our result elaborates the criterion given by Krein and Langer in their joint paper of 1977: they overlooked one attainable case. The correct condition was stated by Langer and Winkler in 1998, although they provided no proper reasoning.