Functions of Regular Growth

Abstract

We have seen in Chapter 3 that there is a relationship between the asymptotic growth of the quantity M _f (r) for an entire function f and the area of the zero set of f. In certain cases, however, much more can be said. We shall prove here the fundamental principle for functions of finite order and of regular growth, which, paraphrased a little crudely, states that an entire function of finite order has its zero set “regularly distributed” asymptotically if and only if it has “regular growth” asymptotically along all rays. An equivalent formulation, as we shall see, is to say that for an entire function f, r^−ρ(r) log ∣f (r z)∣ converges (as a distribution) in L ¹_loc (ℂⁿ) to h ^⋆ _f (z) if and only if r^−ρ(r)Δlog ∣ f(r z)∣ converges as a distribution to Δh ^⋆ _f (z). These ideas of regularity will be made more precise below.