ABSTRACT
As is known, a great part of the theory of functions of a complex variable extends to vector-valued functions, but there exist cases in which the extension to this general setting is no longer valid. Such an example is given by Hille and Phillips (1957) in connection with the maximum principle. In recent years, in connection with some problems in operator theory, interesting results were obtained in connection with some numerical functions associated with an operator-valued function. The aim of this chapter is to give an account of the results obtained.
