Algebroid Functions Research Articles

Algebroid functions, Wirsing's theorem and their relations

In this paper, we first point out a relationship between the Second Main Theorem for algebriod functions in Nevanlinna theory and Wirsing's theorem in Diophantine approximation. This motivates a unified proof for both theorems. The second part of this paper deals with “moving targets” problem for holomorphic maps to Riemann surfaces. Its counterpart in Diophantine approximation follows from a recent work of Thomas J. Tucker. In this paper, we point out Tucker's result in the special case of the approximation by rational points could be obtained by doing a “translation” and applying the corresponding result with fixed target. However, we could not completely recover Tucker's result concerning the approximation by algebraic points. In the last part of this paper, cases in higher dimensions are studied. Some partial results in higher dimensions are obtained and some conjectures are raised.

Monomials of Entire Algebroid Functions

Hayman has shown that iffis a transcendental entire function andn≥2, thenfnf′ assumes all values except possibly zero infinitely often. We extend his result in three directions by considering an entire algebroid functionw, its monomialwn0(w′)n1···(w(k))nk, and by estimating the growth of the number ofa-points of the monomial.

ON ALGEBROID FUNCTIONS IN A SUBDOMAIN OF THE COMPLEX PLANE

This paper discusses some basic properties of algebroid functions in a subdomain of the complex plane and some similar results (to those in the whole complex plane) and a new situation are showed.

Value distribution of some monomials of entire algebroid functions*

We obtain the following result Let w(z) be a v-valued transcendental entire algebroid function and set Then if n0≥4v−2+2(v−1)(n1−1), we have for eace , where p:=n0−4v+3−2(v−1)(n1−1)≥1.

Value distribution of certain monomials of algebroid functions

AbstractHayman has shown that iffis a transcendental meromorphic function andn≽ 3, thenfnf′ assumes all finite values except possibly zero infinitely often. We extend his result in three directions by considering an algebroid function ω, its monomial ωn0ω′n1, and by estimating the growth of the number of α-points of the monomial.

Value distribution of certain differential polynomials of algebroid functions

Value distribution of certain differential polynomials of algebroid functions

Value distribution of some differential polynomials of entire algebroid functions

We obtain the following results: (i) Let w( z) be a v-valued transcendental entire algebroid function and set where and . Then if n ≥ 4v - 1, we have for each (ii) Let n(z) be are valued transcende...

Hayman-Miles Theorem and 2-Valued Algebroid Functions

Given an algebroid function w satisfying $$w^\nu+A_{\nu-1}(z)w^{\nu-1}+\cdot\cdot\cdot+A_0(z)=0,$$ we establish two general methods to calculate the corresponding coefficients in the defining equation for w′ in terms of the meromorphic functions A0, …, Aν−1 and their derivatives. We then obtain the following result: Let w be a 2-valued algebroid function. Then T(r,w′) = o(T(r,w)) can hold at most on a set of zero upper logarithmic density.

Uniqueness theorems for rational, algebraic, and algebroid functions

A number of pointsA, for which one must define the sets of simpleA-points to determine a rational function, a polynomial, and an algebraic or algebroid function uniquely, is obtained.

MAIN THEOREM ON COVERING SURFACES

In this paper, we introduce the concepts of cut ratio and covered ratio of domains. Further, we give an explicit estimation of the constant in the general Ahlfors’ covering theorem and Ahlfors'fundamental theorem on a sphere. Because of the importance of these theorems, it is no doubt that these rstimations is significant. Applying these yesults, we improve a series of theorems, prove that there exists a sequence of filling circles for algebroidal function, and establish a new normal criterion. (These works will be done in other papers.)

Defect relations of holomorphic curves of lower order less than one

Many years ago, Edrei and Fuchs established an elliptic theorem and so solved completely the problem of finding relations between any two deficiencies of a meromorphic function of order less than one. In this paper, an elliptic theorem for a holomorphic curve x:C→P n Cof lower order less than one is established. Particularly an analogue of result of Sato's for algebroid functions is derived.

Further results on the deficiencies of algebroid functions

Let f(z) be an n-valued algebroid function of finite lower order μ. In this paper, we give some further results on the deficiencies of f(z). Particularly if 0 ≤ μ ≤ 1/2, the corresponding result is best possible.

The deficiencies of algebroid functions of order less than one

The deficiencies of algebroid functions of order less than one

Sums of deficiencies of algebroid functions

Let f(z) be an n–Valued algebroid function of finite lower order. In the present paper, we give a spread relation of f(z) and some applications of the spread relation.

On an extension of a theorem of Sato

Let f(z) be a n-valued algebroid function of order λ(0 ≤ λ ≤ 1), Sato obtained an elliptic theorem for f(z) with a condition. In this paper, we prove that Sato's theorem is true without conditions and give a generalisation.

On algebroid functions taking the same values at the same points

On algebroid functions taking the same values at the same points

Uniqueness of the solution of the boundary-value problem with an integral condition for a differential equation in a strip

I. A. A. Gol'dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions [in Russian], Nauka, Moscow (1970). 2. A. Z. Mokhon'ko, "On differential and functional equations with multipliers," Differents. Uravn., 15, No. 9, 1713-1715 (1979). 3. V. D. }~khon'ko, "On the meromorphic solutions of a class of functional equations," Ukr. Mat. Zh., 34, No. 2, 222-225 (1982). 4. A. Z. Mokhon'ko, "On a class of differential and functional equations," Differents. Uravn., 19, No. 3, 414-419 (1983). 5. J. Valiron, Analytic Functions [Russian translation], Gosudarstvennoe Izd. Fizikomatematicheskoi Literatury, Moscow (1957). 6. R. Goldstein, "On meromorphic solutions of certain functional equations," Aequat. Math., 1._8.8, I12-157 (1978). 7. J. M. Baker and F. Gross, "On factorizing entire functions," Proc. London Math. Sot., 3, No. I, 69-76 (1968). 8. B. L. Van der Waerden, Algebra [Russian translation], Nauka, ~scow (1976). 9. A. Z. Mokhon'ko, "The Nevanlinna characteristics of the composition of rational and algebroidal functions,".. Ukr. Mat. Zh., 34, No. 3, 388-396 (1982). 10. H. L. Selberg, "Uber die Wertverteilung der algebroiden Funktionen," Math. Z., 31, 709728 (1930). 11. A. Z. Hokhon'ko, "On the Nevanlinna characteristics for a class of meromorphic curves," Teor. Funktsii, Funkts. Anal. Prilozhen., 25, 95-105 (1976). 12. A. E. Eremenko, "Meromorphic solutions of algebraic differential equations," Usp. Hat. Nauk, 37, No. 4 (226), 53-82 (1982).

Nevanlinna chaeacteristics of a composition of rational and algebroid functions

Nevanlinna chaeacteristics of a composition of rational and algebroid functions

On the asymptotic behaviour of algebroid functions of extremal growth

On the asymptotic behaviour of algebroid functions of extremal growth

Estimates of Nevanlinna characteristics of algebroid functions and their applications to differential equations

Estimates of Nevanlinna characteristics of algebroid functions and their applications to differential equations