Monotonicity properties of the ratio $$\begin{aligned} \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, \end{aligned}$$where f is an entire function are investigated. Earlier results for Euler’s gamma and other functions genus 1 generalised to p with negative zeros. Derivatives order comparable expression above related Stieltjes \(p+1\). Our applied Barnes multiple functions. We al...

In this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. We obtain one results, which can give a negative answer to an uniqueness question relate to the Bruck conjecture dealt by Liu and Yang. Meanwhile, we also establish a difference analogue of the Bruck conjecture for entire functions of order less than 2, ...

it is immaterial which value of z is used in (2). If (1) holds in a region of the s-plane, for example in an angle, ƒ(z) is said to be of exponential type c in that region. Functions of exponential type have been extensively studied, both for their own sake and for their applications. I shall discuss here a selection of their properties, chosen to illustrate how the restriction (1) on the growt...

Since this series is absolutely convergent everywhere in the plane, lanl must approach zero as n approaches infinity. Consequently, there exists for each a, an index n(a) for which lanl is a maximal coefficient. B. Lepson [3]1 raised the question of characterizing entire functions for whidi n (a) is bounded in a. 2 In the sequel we shall consider certain interesting variations of Lepson's probl...

in this paper, we mainly investigate the uniqueness of the entire function sharing a small entire function with its high difference operators. we obtain one results, which can give a negative answer to an uniqueness question relate to the bruck conjecture dealt by liu and yang. meanwhile, we also establish a difference analogue of the bruck conjecture for entire functions of order less than 2, ...

Let G(k) = ∫ 1 0 g(x)e kxdx, g ∈ L1(0, 1). The main result of this paper is the following theorem. THEOREM 1. There exists g 6≡ 0, g ∈ C∞ 0 (0, 1), such that G(kj) = 0, kj < kj+1, limj→∞ kj =∞, limk→∞ |G(k)| does not exist, lim supk→+∞ |G(k)| = ∞. This g oscillates infinitely often in any interval [1− δ, 1], however small δ > 0 is. MSC: 30D15, 42A38, 42A63

Since this series is absolutely convergent everywhere in the plane, the terms \an\ must approach 0. Consequently, there exists for each a, an index n0 = n(a) ior which \an\ is a maximal coefficient. B. Lepson [2] raised the problem of characterizing entire functions for which n(a) is bounded. The latter are called functions of bounded index. In what follows, we shall give a partial solution to ...