Borel Directions Research Articles

On new singular directions of some Schröder functions

In this paper, we study the Hayman T directions and the precise Borel directions of maximal kind of meromorphic solutions f(z) of the Schroder equations f(sz)=R(f(z)), where |s|<1 and R(w) is a rational function with $\deg [R]\geq 2$. We will show that, if $\operatorname {arg}[s]/2\pi \notin Q$, then f(z) has any direction as Hayman T direction and maximus Borel direction as well. This is a continue work of [Ishizaki, K. and Yanaihara, N., Borel and Julia directions of meromorphic Schroder functions, Math. Proc. Camb. Phil. Soc. 139 (2005), 139-147.] and [Yuan, W.J., Qi, J.M. and Seiki Mori. Singular directions of meromorphic solutions of some non-autonomous Schroder equations, Complex Analysis and its Applications Proceedings of the 15th ICFIDCAA held in Osaka (Japan), July 30-August 3, 2007].

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

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.

ANGULAR DISTRIBUTION OF SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this paper, we study the location of zeros and Borel direction for the solutions of linear homogeneous differential equations $$f^{(n)}+A_{n-1}(z)f^{(n-1)}+{\cdots}+A_1(z)f#+A_0(z)f

Borel direction relative to function-values of meromorphic functions with finite logarithmic order

It is shown that if $ f(z )$ is meromorphic in the complex plane $ \mathbb C $ with finite positive logarithmic order $ \lambda $ and its characteristic function $ T(r,f) $ satisfies the growth condition $$ \ls \ T(r,f)/(\log r)^2 = + \infty,$$ then there is a number $ \theta $ with $ 0 \le \theta &lt; 2\pi $ such that for each positive number $ \epsilon $, the expression $$ \ls \ \dfrac{\log \bigg\{\displaystyle{\sum^3_{i=1}} \ n(r,\theta, \epsilon, f = a_i(z)) \bigg\}}{\log \log r} = \lambda - 1, $$ holds for any three distinct meromorphic functions $ a_i(z) (i = 1,2,3) $ with $ T(r,a_i) = o(U(r,f)/ $ $ (\log r)^2), $ as $ r \to + \infty $, where $ n(r,\varphi ,\epsilon ,f = a(z)) $ denotes the number of roots counting multiplicitie s of the equation $ f(z) = a(z) $ for $ z$ in the angular domain $ \Omega (r,\varphi ,\epsilon ) = \{z: |\arg z - \varphi | &lt; \epsilon $, $ |z| &lt; r \} $ where $ 0 \leq \varphi &lt; 2\pi $, $ \epsilon &gt; 0$, $U(r,f) = (\log r)^{\lambda (r)} $, and $ \displaystyle{\ls \ \lambda (r) = \lambda} $.

The Borel direction of the largest type of algebroid functions dealing with multiple values

The Borel direction of the largest type of algebroid functions dealing with multiple values

Borel and Julia directions of meromorphic Schröder functions II

Let R(w) be a non-inear rational function and s be a complex constant with | s | > 1. It is showed that for any solution f (z) of the Schroder equation f (sz) = R(f (z)), Julia directions of f (z) are also Borel directions of f (z).

ON NEVANLINNA DIRECTIONS OF ALGEBROID FUNCTIONS

In this paper, a notation Δχ(ω) is derived from the counting function Nχ(r, ω) of branch points of algebriod functions. With this notation, the authors give the definition of the Nevanlinna direction for algebriod functions and discuss its existence in certain condition. By this notation the authors also obtain the numbers of exceptional value of the Julia direction and Borel direction of algebriod functions are not more than 2 + [Δχ(ω)], here [x] implies an maximum integer number which does not exceed x.

On a Singular Direction of a Meromorphic Function

By using Ahlfors' theory of a covering surface, we establish the existence of a new singular direction for a meromorphic function 𝑓, namely a 𝑇 direction for 𝑓, for which the Nevanlinna characteristic function 𝑇(𝑟, 𝑓) is used as a comparison function. Then we prove that every 𝑇 direction is a Borel direction for meromorphic function with finite and positive regular growth order.

Angular distribution in complex oscillation theory

Let f 1 and f 2 be two linearly independent solutions of the differential equation f ″ + Af = 0, where A is an entire function. Set E = f 1 f 2 . In this paper, we shall study the angular distribution of E and establish a relation between zero accumulation rays and Borel directions of E . Consequently we can obtain some results in the complex differential equation by using known results in angular distribution theory of meromorphic functions.

ON THE BOREL DIRECTION OF K-QUASIMEROMORPHIC MAPPINGS

For the K-quasimeromorphic mappings, a precise fundamental inequality for the angular domain is established. From this, the Borel direction of the K-quasimeromorphic mappings of zero order is derived.

Common Borel directions of a meromorphic function with zero order and its derivative

There is a meromorphic function of zero order for which the function and its derivative have no common Borel direction.

SOME REMARKS ON QUASIMEROMORPHIC MAPPINGS

In this paper, the Nevanlinna direction, Borel direction and exceptional value of quasimeromorphic mapping are discussed and some results are obtained.

On the Value Distribution of an Algebroid Function of Infinite Order

In this paper, we first give a fundamental inequality of algebroid functions, then using it we prove that for any x -valued algebroid functions of infinite order in the plane, there exists a sequence of filling disks and Borel direction of dealing with its multiple values, such that the number of Borel exceptional values is equal to 2ν at most.

On the distribution of values and deficient function of a general random integral function

Under suitable conditions on {Xn}, the author obtains the important results: it is almost sure that the random integral function fω=∑n=0∞Xnzn (of finite positive order) has no deficient function, and any direction is a Borel direction (without finite exceptional value) of fω.

Borel directions and iterated orbits of meromorphic functions

For transcendental meromorphic functions of finite order, we prove that there exist iterated orbits which tend to the Borel directions. This gives a relation between the value distribution theory and the iteration theory of meromorphic functions.

Value distribution of quasiconformal mappings*

For the K-quasiconformal mappings in the complex plane, a precise fundamental inequality is established. From this, the defect relation, filling disks, Borel directions and normal criterion are also derived.

A fundamental inequality for meromorphic functions in an angular domain and its application

This paper is to study the characteristic functions of meromorphic functions for an angular domain. We show an inequality which is similar to the inequality in the second fundamental theorem of Nevanlinna, and estimate its remainder. Finally, as an application we obtain a sufficient and necessary condition of Borel directions.

Angular distribution and deficient values of entire functions and their derivatives

Let f be an entire function of positive and finite lower order are q (1 ≤ q < + ∞) rays issued from the origin. If f and f (l) (l = 1, 2 …) have no Borel direction of order ≥ μ except for , then the sum of the numbers of the finite nonzero deficient values off and all its primitives and derivatives is smaller than 2μ. The proof is based on several lemmas and application of harmonic measure.

Deficient values and angular distribution of entire functions

Let f ( z ) f(z) be an entire function of positive and finite order μ \mu . If f ( z ) f(z) has a finite number of Borel directions of order ⩾ μ \geqslant \mu , then the sum of numbers of finite nonzero deficient values of f ( z ) f(z) and all its primitives does not exceed 2 μ 2\mu . The proof is based on several lemmas and application of harmonic measure.

ANGULAR DISTRIBUTION AND MULTIPLE.VALUES BETWEEN ENTIRE FUNCTIONS AND THEIR DERIVATIVES

The present paper is concerned with an important relation between Borel directions and multiple values of entire functions of order λ (0<λ<∞) and their derivatives.