Global and local behavior of zeros of nonpositive type

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