On Zeros of Holomorphic Functions

Abstract

In this case the minimum k0 is equal to the order of function. Recall that the entire function f(z) has a finite order (of growth) if there exists a positive number A such that f(z) = O(e A ) for |z| = R → +∞. The infimum of such numbers A is called the order of function (see, e.g., [2, 3]). Proof. Let the function f be a function of finite order of growth, which has no zeros in C then it is well known that it has the form: f(z) = e, where φ(z) is a polynomial of some degree k0 (see, e.g., [2, Ch. 7, Sec. 1.5]). Then ∫