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))Qτ(z)=(Q(z)−τ)/(1+τQ(z)), τ∈R∪{∞}τ∈R∪{∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ)α(τ) 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.