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On orders and types of Dirichlet series of slow growth

The present paper has the object of showing some interesting relationship on the maximum modulus, the maximum term, the index of maximum term and the coefficients of entire functions defined by Dirichlet series of slow growth some properties like Taylor entire functions are obtained. | Turk J Math 34 (2010) , 1 – 11. ¨ ITAK ˙ c TUB doi:10.3906/mat-0810-17 On orders and types of Dirichlet series of slow growth Yinying Kong and Huilin Gan Abstract The present paper has the object of showing some interesting relationship on the maximum modulus, the maximum term, the index of maximum term and the coefficients of entire functions defined by Dirichlet series of slow growth; some properties like Taylor entire functions are obtained. Key Words: Dirichlet series, generalized order, generalized type. 1. Introduction and main results The growth and the value distribution of Taylor entire functions f(z) = +∞ bn z n n=0 were studied for a long time and many important results were obtained in [1],[2] and [3]. For instance, S.K. Bajpai gave some different characterizations on the coefficients and the maximum modulus, the maximum term, and the index of maximum term for the entire functions of fast growth ρ = ∞ in [1] . On the other hand, G.P. Kapoor [3] and Ramesh Ganti [2] continued this work and defined a generalized order and a generalized type for the Taylor entire functions of slow growth ρ = 0 . Dirichlet series was introduced by L. Dirichlet in 19th century and it has the form: f(s) = +∞ bn eλn s , (1) n=1 where {bn } ∈ C, 0 0 , that is, h(x) is slowly increasing. Definition 2 Let α(x) ∈ Λ , the generalized order of the entire function f(s) defined by (1) can be defined as α(ln M (σ)) , σ→+∞ α(σ) ρ = ρ(α; f) = lim 2 KONG, GAN if the order is of slow growth i.e. ρ ∈ (0, ∞), and then the type τ (α; f) of (1) is defined by τ = τ (α; f) = lim σ→+∞ α(M (σ)) β(ln M (σ)) = lim , σ ρ σ→+∞ [α(e )] [β(σ)]ρ where β(ln x) = α(x). Theorem 1 Suppose that Dirichlet series (1) satisfies (2) and (3), then 1o 2o lim lim σ→+∞ α(ln M (σ)) α(λn ) − 1 = lim , 1 σ→+∞ α(σ) α(ln |bn |− λn ) α(λn ) σ→+∞ α(ln |b n| − λ1n ) ≤ lim σ→+∞ for p = 1, α(ln M (σ)) α(λn ) + 1, ≤ lim 1 σ→+∞ α(σ) α(ln |bn |− λn ) for p = 2, 3, · · · Theorem 2 Suppose that .