Properties of Certain Spaces of Entire Functions

Minaketan Das
1968 Proceedings of the American Mathematical Society  
1. A. C. Offord [3] has discussed the behaviour of Frechet's space 5 of entire functions. He has shown that "most" (viz. those belonging to a set whose complement is of the first category in ff) entire functions behave in a rather wild manner. In this note we show that "most" entire functions of ff (i) are of lower order zero, (ii) possess certain properties which are ordinarily met with the polynomials "only" (cf. (1) , (2) and (3) below) . We mention here some of the properties of the space
more » ... ties of the space J, viz: (a) A sequence of entire functions is convergent if it converges uniformly on every closed bounded set of the complex plane. (b) JF is a complete metric space. (c) The set of all polynomials is dense in 5. It follows from (b) that ff is of the second category. The properties (a), (b) and (c) happens to be possessed by the space T of all entire functions, considered by V. Ganapathy Iyer in [2]. Our methods are variants of those of Offord, who like us, uses only the properties (a)-(c) mentioned above. As such all his results and those below hold for Iyer's space too. We denote the number of zeros of a function/(z) in the open region \z\ <r by w*(r,/); also N denotes a fixed positive integer. We prove Theorem. Let n, r2, rz, ■ ■ ■ be an increasing and unbounded set of positive constants. Then there is a set ff* of entire functions whose comple-
doi:10.2307/2035306 fatcat:43r2wonjbjh67cffpnvd4ysw7y