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Hyperbolicity plays an important role in the study of dynamical systems, andis a key concept in the iteration of rational functions of one complexvariable. Hyperbolic systems have also been considered in the study oftranscendental entire functions. There does not appear to be an agreeddefinition of the concept in this context, due to complications arising fromthe non-compactness of the phase space.  In this article, we consider a natural definition of hyperbolicity thatrequires expanding properties on the preimage of a punctured neighbourhood ofthe isolated singularity. We show that this definition is equivalent to anothercommonly used one: a {\tef} is hyperbolic if and only if its postsingular setis a compact subset of the Fatou set. This leads us to propose that this notionshould be used as the general definition of hyperbolicity in the context ofentire functions, and, in particular, that speaking about hyperbolicity makessense only within the \emph{Eremenko-Lyubich class} $\mathcal{B}$ oftranscendental entire functions with a bounded set of singular values.  We also considerably strengthen a recent characterisation of the class$\mathcal{B}$, by showing that functions outside of this class cannot beexpanding with respect to a metric whose density decays at most polynomially.In particular, this implies that no transcendental entire function can beexpanding with respect to the spherical metric. Finally we give acharacterisation of an analogous class of functions analytic in a hyperbolicdomain.

Hyperbolic entire functions and the Eremenko-Lyubich class: Class $\mathcal{B}$ or not class $\mathcal{B}$?