By Stephen D. Fisher and Jonathan E. Shapiro

This paper appears in the Illinois Journal of Mathematics, 43 (1999),
113-130.

**Abstract:**

We generalize to finitely connected planar domains the result of Joel
Shapiro which gives a formula for the essential norm of a composition operator.
In the process, we define and give some properties of a generalization
of the Nevanlinna counting function and prove generalizations of the Littlewood
inequality, the Littlewood-Paley identity, and change of variable formulas,
as well.

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