scholarly journals ON BOREL DIRECTION CONCERNING SMALL FUNCTIONS

2021 ◽  
Vol 29 (1) ◽  
pp. 13-16
Author(s):  
T. Y. PETER CHERN
Keyword(s):  
Meromorphic Function ◽  
Proximate Order ◽  
Positive Order ◽  
Borel Direction ◽  
Prove Theorem ◽  
Small Functions

In this paper, we shall prove Theorem 1 Let $f$ be nonconstant meromorphic  in $\mathbb{C}$ with finite positive order $\lambda$, $\lambda(r)$ be a proximate order of $f$ and $U(r, f)=r^{\lambda(r)}$, then for each number $\alpha$,$0<\alpha<\pi/2$, there exists a number $\phi_0$ with $0\le \phi_0 < 2\pi$ such that the inequality \[ \limsup_{r\to\infty}\sum_{i=1}^3 n(r, \phi_0, \alpha, f=a_i(z))/U(r, f)>0,\] holds for any three distinct meromorphic function $a_i(z)(i=1, 2, 3)$ with $T(r,a_i)=o(U(r, f))$ as $r\to\infty$.

scholarly journals Derivatives of Meromorphic Functions with Multiple Zeros and Small Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Pai Yang ◽  
Peiyan Niu
Keyword(s):  
Meromorphic Function ◽  
Rational Function ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Infinitely Many Solutions ◽  
Multiple Zeros ◽  
Small Functions ◽  
Derivatives Of

Letfzbe a meromorphic function inℂ, and letαz=Rzhz≢0, wherehzis a nonconstant elliptic function andRzis a rational function. Suppose that all zeros offzare multiple except finitely many andTr,α=oTr,fasr→∞. Thenf'z=αzhas infinitely many solutions.

scholarly journals On the Uniqueness of Power of a Meromorphic Function Sharing a set with its k-Th Derivative

2018 ◽  
Vol 85 (1-2) ◽  
pp. 1
Author(s):  
Abhijit Banerjee ◽  
Bikash Chakraborty
Keyword(s):  
Meromorphic Function ◽  
Small Functions ◽  
Set Sharing ◽  
Function Sharing

<p>In the existing literature, many researchers consider the uniqueness of the power of a meromorphic function with its derivative counterpart share certain values or small functions. Here we consider the same problem under the aegis of a more general settings namely set sharing.</p>

T-Direction and Borel Direction of Algebroid Functions of Finite and Positive Order

2009 ◽  
Vol 16 (3) ◽  
pp. 583-596
Author(s):  
Zhao-Jun Wu
Keyword(s):  
Positive Order ◽  
Algebroid Function ◽  
Borel Direction ◽  
Algebroid Functions

Abstract In the present paper, the connection between a T-direction and Borel direction of algebroid functions is investigated. Two algebroid functions (not meromorphic) are obtained to prove that T-directions and Borel directions are two different classes of singular directions for algebroid functions. The existence of an algebroid function (not meromorphic) with a given Borel direction is proved.

scholarly journals On the value distribution of a differential monomial and some normality criteria

2021 ◽  
Vol 56 (1) ◽  
pp. 55-60
Author(s):  
W. Lü ◽  
B. CHAKRABORTY
Keyword(s):  
Meromorphic Function ◽  
Analytic Functions ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Zero Distribution ◽  
Small Functions ◽  
Transcendental Meromorphic Function ◽  
Distribution Result ◽  
Positive Integers

The aim of this paper is to study the zero distribution of the differential polynomial $\displaystyle af^{q_{0}}(f')^{q_{1}}...(f^{(k)})^{q_{k}}-\varphi,$where $f$ is a transcendental meromorphic function and $a=a(z)(\not\equiv 0,\infty)$ and $\varphi(\not\equiv 0,\infty)$ are small functions of $f$. Moreover, using this value distribution result, we prove the following normality criterion for family of analytic functions:\\ {\it Let $\mathscr{F}$ be a family of analytic functions on a domain $D$ and let $k \geq1$, $q_{0}\geq 2$, $q_{i} \geq 0$ $(i=1,2,\ldots,k-1)$, $q_{k}\geq 1$ be positive integers. If for each $f\in \mathscr{F}$: i.\ $f$ has only zeros of multiplicity at least $k$,\ ii.\ $\displaystyle f^{q_{0}}(f')^{q_{1}}\ldots(f^{(k)})^{q_{k}}\not=1$,then $\mathscr{F}$ is normal on domain $D$.

scholarly journals Value Distribution for a Class of Small Functions in the Unit Disk

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Paul A. Gunsul
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Unit Disk ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Finite Measure ◽  
Differential Polynomials ◽  
Proximity Function ◽  
Small Functions ◽  
Linear Differential

If is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function could be used to categorize according to its rate of growth as . Later H. Milloux showed for a transcendental meromorphic function in the plane that for each positive integer , as , possibly outside a set of finite measure where denotes the proximity function of Nevanlinna theory. If is a meromorphic function in the unit disk , analogous results to the previous equation exist when . In this paper, we consider the class of meromorphic functions in for which , , and as . We explore characteristics of the class and some places where functions in the class behave in a significantly different manner than those for which holds. We also explore connections between the class and linear differential equations and values of differential polynomials and give an analogue to Nevanlinna's five-value theorem.

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