Nonconstant Entire Function Research Articles

Unicity on Entire Functions Concerning Their Difference Operators and Derivatives

In this paper we study the uniqueness of entire functions concerning their difference operator and derivatives. The idea of entire and meromorphic functions relies heavily on this direction. Rubel and Yang considered the uniqueness of entire function and its derivative and proved that if $f(z)$ and $f'(z)$ share two values $a,b$ counting multilicities then $f(z)\equiv f'(z)$. Later, Li Ping and Yang improved the result given by Rubel and Yang and proved that if $f(z)$ is a non-constant entire function and $a,b$ are two finite distinct complex values and if $f(z)$ and $f^{(k)}(z)$ share $a$ counting multiplicities and $b$ ignoring multiplicities then $f(z)\equiv f^{(k)}(z)$. In recent years, the value distribution of meromorphic functions of finite order with respect to difference analogue has become a subject of interest. By replacing finite distinct complex values by polynomials, we prove the following result: Let $\Delta f(z)$ be trancendental entire functions of finite order, $ k \geq 0$ be integer and $P_{1}$ and $P_{2}$ be two polynomials. If $\Delta f(z)$ and $f^{(k)}$ share $P_{1}$ CM and share $P_{2}$ IM, then $\Delta f \equiv f^{(k)}$. A non-trivial proof of this result uses Nevanlinna's value distribution theory.

Uniform Convergence of Translation Operators

We denote N, R, C the sets of natural, real and complex numbers respectively. Let (λn), n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function f with the following property: for every x, y ∈ R with 0 , every θ ∈(0,1) and every a ∈ C there is a subsequence of natural numbers (mn), n ∈ N such that, for every compact subset L ⊆ C , In the present paper we show that the constant function a cannot be replaced by any non-constant entire function G. This is so even if one demands the convergence in (*) only for a single radius r and a single positive number θ. This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.

ENTIRE FUNCTIONS THAT SHARE A SET WITH THEIR DIFFERENCES

<abstract> In this paper, we study the uniqueness of entire functions concerning deficient value and exponent of convergence, and have mainly proved the following theorem: Let $S=\{1, \omega , \omega ^2, \cdots , \omega ^{n-1}\}$, where $\omega ^n=1$, $n\ge 1$ is an integer, let $k$ be a positive integer, and let $f$ be a nonconstant entire function such that $\lambda(f)&lt;\rho(f)&lt;\infty$. If $f(z)$ and $\Delta _{\eta }^kf(z)$ share $S$ IM, where $\eta $ is a nonzero complex number, then $f(z)=e^{az+b}$, where $a(\neq0)$ and $b$ are two finite complex numbers. The results obtained in this paper improve some results due to Li ([<xref ref-type="bibr" rid="b15">15</xref>]). </abstract>

Entire functions that share two pairs of small functions

AbstractIn this paper, we study the unicity of entire functions and their derivatives and obtain the following result: letffbe a non-constant entire function, leta1{a}_{1},a2{a}_{2},b1{b}_{1}, andb2{b}_{2}be four small functions offfsuch thata1≢b1{a}_{1}\not\equiv {b}_{1},a2≢b2{a}_{2}\not\equiv {b}_{2}, and none of them is identically equal to∞\infty. Ifffandf(k){f}^{\left(k)}share(a1,a2)\left({a}_{1},{a}_{2})CM and share(b1,b2)\left({b}_{1},{b}_{2})IM, then(a2−b2)f−(a1−b1)f(k)≡a2b1−a1b2\left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2}. This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan.51(1999), no. 7, 781–799].

Relative logarithmic order of an entire function

In this paper, we extend some results related to the growth rates of entire functions by introducing the relative logarithmic order $\rho _{g}^{l}(f)$ of a nonconstant entire function $f$ with respect to another nonconstant entire function $g.$ Next we investigate some theorems related the behavior of $\rho _{g}^{l}(f)$. We also define the relative logarithmic$ $proximate order of $f$ with respect to $g$ and give some theorems on it.

Unicity of Entire Functions Concerning Their Shifts and Derivatives

In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If $$f'(z)$$ and $$f(z+c)$$ share a, b IM, then $$f'(z)\equiv f(z+c)$$ . This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178, 2020).

Further study on the Brück conjecture and some non-linear complex differential equations

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramaniket al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex numberain the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Letfbe a non-constant entire function such thatσ2(f)&lt;∞,σ2(f)is not a positive integer andδ(0, f)&gt;0. LetM[f]be a differential monomial offof degreeγMandα(z), β(z)∈S(f)be such thatmax{σ(α), σ(β)} &lt;σ(f). IfM[f]+βandfγM−αshare the value 0 CM, thenM[f]+βfγM−α=c,wherec≠0is a constant.Originality/valueThis is an original work of the authors.

Relative $L$-order of an entire function

In this paper we introduce relative $L-$order of a nonconstant entire function $f$ with respect to another nonconstant entire function $g$. Also we investigate the existence of relative $L-$proximate order of $f$ with respect to $g$.

Some unity results on entire functions and their difference operators related to 4 CM theorem

This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let f(z) be a nonconstant entire function of finite order, and let a_{1}, a_{2} be two distinct finite complex constants. If f(z) and Delta _{eta }^{n}f(z) share a_{1} and a_{2} “CM”, then f(z)equiv Delta _{eta }^{n} f(z), and hence f(z) and Delta _{eta }^{n}f(z) share a_{1} and a_{2} CM.

Meromorphic Functions Sharing four Values with their Difference Operators

In 2011, Heittokangas et al. (Complex Var Ellipt Equat 56(1–4):81–92, 2011) proved that if a non-constant finite order entire function f(z) and $$f(z+\eta )$$ share a, b, c IM, where $$\eta $$ is a finite non-zero complex number, while a, b, c are three distinct finite complex values, then $$f(z)=f(z+\eta )$$ for all $$z\in \mathbb {C}$$ . We prove that if a non-constant finite order entire function f and its n-th difference operator $$\Delta ^n_{\eta }f$$ share $$a_1$$ , $$a_2$$ , $$a_3$$ IM, where n is a positive integer, $$\eta \ne 0$$ is a finite complex value, while $$a_1$$ , $$a_2$$ , $$a_3$$ are three distinct finite complex values, then $$f=\Delta ^n_{\eta }f$$ . The main results in this paper also improve Theorems 1.1 and 1.2 from Li and Yi (Bull Korean Math Soc 53(4):1213–1235, 2016).

Some results on entire functions that share one value with their difference operators

In this paper, we give some results on entire functions that share one value with their difference operators. In particular, we prove the following result, which can be regarded as a difference analogue of a result of J.P. Wang and H.X. Yi (J. Math. Anal. Appl. 277:155–163, 2003): Let f(z) be a non-constant entire function such that rho_{2}(f)<1, a (neq0) be a finite constant, and n and m be positive integers satisfying m>n>1. If \t\t\tf(z)=a⇌Δcf(z)=a,f(z)=a→Δcmf(z)=Δcnf(z)=a,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ f(z)=a \\rightleftharpoons\\Delta_{c}f(z)=a,\\qquad f(z)=a \\rightarrow \\Delta_{c}^{m} f(z)=\\Delta_{c}^{n} f(z)=a, $$\\end{document} then Delta_{c}^{n}f(z)equivDelta_{c}^{m}f(z). Two related results are proved and an example is provided.

Entire functions sharing one small function CM with their shifts and difference operators

In this article, we mainly devote to proving uniqueness results for entire functions sharing one small function CM with their shift and difference operator simultaneously. Let f(z) be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f(z), f(z+c), and Δcnf(z) share 0 CM, then f(z + c)≡ Af(z), where A(≠ 0) is a complex constant. Moreover, let a(z), b(z)(≢ 0) ∈ S(f) be periodic entire functions with period f(z)−a(z), f(z+c)−a(z), Δcnf(z)−b(z) share 0 CM, then f(z + c) ≡ f(z).

On the study of Brück conjecture and some non-linear complex differential equations

In this paper, we prove the following result: Let f(z) and α(z) be two non-constant entire functions satisfying σ(α)<μ(f) and P(z) be a polynomial. If f is a non-constant entire solution of the differential equation M[f]+β(z)−α(z)=(fγM−α(z))eP(z), where β(z) is an entire function satisfying σ(β)<μ(f). Then σ2(f)=degP. Our result generalizes the results due to Gundersen and Yang, Chang and Zhu and Li and Cao.

MEROMORPHIC FUNCTIONS SHARING FOUR VALUES WITH THEIR DIFFERENCE OPERATORS OR SHIFTS

We prove a uniqueness theorem of nonconstant meromorphic functions sharing three distinct values IM and a fourth value CM with their shifts, and prove a uniqueness theorem of nonconstant entire functions sharing two distinct small functions IM with their shifts, which respectively improve Corollary 3.3(a) and Corollary 2.2(a) from [12], where the meromorphic functions and the entire functions are of hyper order less than 1. An example is provided to show that the above results are the best possible. We also prove two uniqueness theorems of nonconstant meromorphic functions sharing four distinct values with their difference operators.

ON THE BRÜCK CONJECTURE

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)&lt;+\infty$ and hyper-order ${\it\sigma}_{2}(f)&lt;{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

Entire functions and their first derivatives sharing simple β-points for a small function β

The following theorem has been proved by A. Schweizer [7]. If a nonconstant entire function f and its derivative f′ share their simple zeros and if every simple a-point of f is a (not necessarily simple) a-point of f′ for some nonzero constant a, then f ≡ f′. In this paper we shall prove that the above result is also true when the nonzero constant a is replaced by a meromorphic small function β( $\not\equiv$ 0, ∞).

A Note on Entire Functions That Share Two Small Functions

This note is to show that iffis a nonconstant entire function that shares two pairs of small functions ignoring multiplicities with its first derivativef', then there exists a close linear relationship betweenfandf'. This result is a generalization of some results obtained by Rubel and Yang, Mues and Steinmetz, Zheng and Wang, and Qiu. Moreover, examples are provided to show that the conditions in the result are sharp.

Entire functions that share a set with their derivatives

There exists a set S with two elements such that if f is a nonconstant entire function, f and f ′ share the set S CM, and if f ′′(z) is nonzero bounded whenever f ∈ S, then f ≡ f ′ ≡ f ′′. This result is a generalization of a result obtained by Jank, Mues and Volkmann.

Meromorphic functions that share five pairs of values

We prove sharp inequalities for two nonconstant meromorphic functions that share five pairs of values ignoring multiplicities (IM). We also obtain the following result: If f and g are nonconstant entire functions that share four finite pairs of values IM, then f is a Möbius transformation of g.

A power of an entire function sharing one value with its derivative

In this paper, we investigate uniqueness problems of entire functions that share one value with one of their derivatives. Let f be a non-constant entire function, n and k be positive integers. If f n and ( f n ) ( k ) share 1 CM and n ≥ k + 1 , then f n = ( f n ) ( k ) , and f assumes the form f ( z ) = c e λ n z , where c is a non-zero constant and λ k = 1 . This result shows that a conjecture given by Brück is true when F = f n , where n ≥ 2 is an integer.