Global and local behavior of zeros of nonpositive type

A generalized Nevanlinna function Q(z)$Q(z)$ with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by _Qτ(z)=(Q(z)−τ)/(1+τQ(z))$Q_{\tau }(z)=(Q(z)-\tau )/(1+\tau Q(z))$, τ∈R∪{∞}$\tau \in \mathbb{R}\cup \{\infty \}$, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ)$\alpha (\tau )$ as a function of τ is being studied. In particular, it is shown that it is continuous and its behavior in the points where the function extends through the real line is investigated.