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Result on an Integral function of an Integral function represented by Dirichlet series

This paper deals in exhibiting a property for which type (R) of an integral function of an integral function represented by Dirichlet series for a finite order (R) is finite.


Introduction
Consider a sequence {a n } of complex numbers and a strictly increasing sequence {λ n } of real numbers such that λ n → ∞ as n → ∞ with λ 1 > 0. Define a function g : C → C such that g(s) = ∞ n=1 a n e λns ; s = σ + it (σ, t ∈ R) (1.1) satisfying lim sup n→∞ log λ n n < ∞ and lim sup n→∞ log |a n | λ n = −∞.
We further define From [1], f(s) and g(s) denote entire functions represented by Dirichlet series in one complex variable 's'. Clearly domain of absolute convergence and convergence of f (s) coincide with each other. The maximum modulus of the entire functions f(s) and g(s) are defined as Ritt [2] defined the order (R) of an entire function g(s) as For 0 < ρ g < ∞, type (R) 'τ g ' of g(s) is defined as We put m(σ, g) = max{|a n |e λnσ : n} as the maximum term of g(s).
K.N. Srivastava in [3] established the conditions under which order (R) of an integral function of an integral function represented by Dirichlet series is finite. In this paper we establish the result for which type (R) of an integral function of an integral function represented by Dirichlet series for a finite order (R) is finite. We take Result on an Integral function of an Integral function 3 2 Basic results K. Sugimura [4] proved the following two lemmas. Then there exists a definite number C not dependent on f(s), g(s) and σ, such that where θ is a fixed constant.
The following result has been proven by Srivastava [3]:

Theorem 2.3 If h(s) is of finite order (R) then one of the following holds a) Either, g(s) is a Dirichlet polynomial and f(s) is of finite order (R); or b) g(s) is of finite order (R) and f(s) is of order (R) zero.
3 Main result and from Lemma 2.2, we have Here, As g(s) is a Dirichlet polynomial, therefore λ m holds a finite value.
Case 2: g(s) is of finite order (R) and f(s) is of order (R) zero.