Treffer: THE THEOREM ON THE LEAST MAJORANT AND ITS APPLICATIONS. II. ENTIRE AND MEROMORPHIC FUNCTIONS OF FINITE ORDER: The theorem of the least majorant and its applications. II: Entire and meromorphic functions of finite order

[Part I cf. ibid. 42, No. 1, 115-131 (1994; Zbl 0798.31003).] Let \(D\geq 0\) be a divisor on \(\mathbb{C}^n\). For an entire function \(f\) on \(\mathbb{C}^n\), the divisor of \(f\) will be denoted by \(D_f'\). It is known that there exists an entire function \(F\) such that \(D_F = D\). For a given positive number \(\rho\) denote by \(\sigma_n (D, \rho)\) the infimum of all the numbers \(\sigma > 0\) such that there exists an entire function \(f\), \(f \equiv 0\) of order \(\rho\), whose type is less than \(\sigma\) and such that \(D_f \geq D\). The aim of this paper is to give lower and upper estimates of \(\sigma_n (D, \rho)\). It is shown that the lower one equals \[ \sigma = \limsup_{t \to \infty} t^{- \rho} \cdot N_F (t) \] where \(N_F (t)\) is the counting function of the zeros of \(F\). As to the sharpness of the upper one, some partial results are given. Applications on the representation of a meromorphic function as a quotient of entire functions and on the completeness of a system of exponents are also given.