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In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region Ω {\displaystyle \Omega } if and only if f {\displaystyle f} is analytic on Ω {\displaystyle \Omega } and log + | f | {\displaystyle \log ^{+}|f|} has a harmonic majorant on Ω , {\displaystyle \Omega ,} where log + = max ] {\displaystyle \log ^{+}=\max]}. Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type , and if Ω {\displaystyle \Omega } is simply connected the condition is also necessary.
The class of all such f {\displaystyle f} on Ω {\displaystyle \Omega } is commonly denoted N {\displaystyle N} and is sometimes called the Nevanlinna class for Ω {\displaystyle \Omega }. The Nevanlinna class includes all the Hardy classes.
Functions of bounded type are not necessarily bounded, nor do they have a property called "type" which is bounded. The reason for the name is probably that when defined on a disc, the Nevanlinna characteristic is bounded.
Clearly, if a function is the ratio of two bounded functions, then it can be expressed as the ratio of two functions which are bounded by 1: