The Growth on the Maximum Modulus of Double Dirichlet Series

Yong-Qin Cui,Hong-Yan Xu,Na Li

doi:10.1155/2019/9191346

Yong-Qin Cui, Hong-Yan Xu + Show 1 more

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https://doi.org/10.1155/2019/9191346

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Abstract

The main purpose of this paper is to investigate the growth of several entire functions represented by double Dirichlet series of finite logarithmic order, h-order. Besides, we also study some properties on the maximum modulus of double Dirichlet series and its partial derivative. Our results are extension and improvement of previous results given by Huo and Liang.

Highlights

Introduction and Basic NotesFor Dirichlet series ∞f (s) = ∑aneλns, s = σ + it, (1) n=1 where0 ≤ λ1 < λ2 < ⋅ ⋅ ⋅ < λn < ⋅ ⋅ ⋅, (2)λn 󳨀→ +∞ as n 󳨀→ +∞;s = σ + it (σ, t are real variables); an are nonzero complex numbers.It is an interesting topic to study some properties of Dirichlet series in the fields of complex analysis;considerable attention has been paid to the analytic function and entire funtcions represented by Dirichlet series in the half-plane and whole plane, and a number of interesting and important results can be found in [1]
The main purpose of this paper is to investigate the growth of several entire functions represented by double Dirichlet series of finite logarithmic order, h-order
We further investigate the growth of entire functions represented by double Dirichlet series, such as the logarithmic order, h-order, and some properties of the maximum modulus of double Dirichlet series and its partial derivatives

Summary

Introduction and Basic Notes

Suppose f(s1, s2) ∈ D; let LT be the set of f(s1, s2) ∈ L given by double Dirichlet series of finite logarithmic order (ρ1L, ρ2L) has finite logarithmic type and f(s1, s2) satisfy the following conditions:. We say that f(s1, s2) has partial logarithmic type τ1L with respect to s1; for any small ε > 0 and fixed value σ1 > 0, there exists σ(2) = σ(2)(ε, σ2) such that. Suppose f(s1, s2) ∈ D and h(x) ∈ F; let LhT be the set of f(s1, s2) ∈ Lh given by double Dirichlet series of finite logarithmic order (ρ1L, ρ2L) having finite h-order, and f(s1, s2) satisfies the following conditions:. We only listed some Liang’s definitions as n = 2; Liang defined the order of multiple Dirichlet series f(s1, s2) by ρ lim sup log log M (σ1, σ2, log (eσ1 + eσ ).

Proofs of Theorems 5–22

Proofs of Theorems 26 and 27

Proofs of Theorems 28 and 29

Conflicts of Interest