The purpose of the paper is to study the uniqueness problems of linear differential polynomials of entire functions sharing a small function and obtain some results which improve and generalize the related results due to J. T. Li and P. Li (2015). Basically we pay our attention to the condition λ(f) ≠ 1 in Theorems 1.3, 1.4 from J. T. Li and P. Li (2015). Some examples have been exhibited to show that conditions used in the paper are sharp.
The spaces of entire functions represented by Dirichlet series have been studied by Hussein and Kamthan and others. In the present paper we consider the space X of all entire functions defined by vector-valued Dirichlet series and study the properties of a sequence space which is defined using the type of an entire function represented by vectorvalued Dirichlet series. The main result concerns with obtaining the nature of the dual space of this sequence space and coefficient multipliers for some classes of vector-valued Dirichlet series.
In this paper, we consider a random entire function $f(s,\omega )$ defined by a random Dirichlet series $\sum \nolimits _{n=1}^{\infty }X_n(\omega ) {\rm e} ^{-\lambda _n s}$ where $X_n$ are independent and complex valued variables, $0\leq \lambda _n \nearrow +\infty $. We prove that under natural conditions, for some random entire functions of order $(R)$ zero $f(s,\omega)$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of $X_n$ for such function $f(s,\omega)$ of order $(R)$ zero, almost surely.
In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let f(z) and g(z) be two transcendental entire functions of finite order, and α(z) a small function with respect to both f(z) and g(z). Suppose that c is a non-zero complex constant and n ≥ 7 (or n ≥ 10) is an integer. If f n (z)(f(z)−1)f(z +c) and g n (z)(g(z) − 1)g(z + c) share ''(α(z), 2)'' (or (α(z), 2)∗ ), then f(z) ≡ g(z). Our results extend and generalize some well known previous results.
In this paper we study the uniqueness for meromorphic functions sharing one value, and obtain some results which improve and generalize the related results due to M. L. Fang, X. Y. Zhang, W. C. Lin, T. D. Zhang, W. R. Lü and others.