This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0013091500003084/resource/name/S0013091500003084_eqnU1.gif?pub-status=live)
For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).