Asymptotic vectors of entire curves

  • Ya.I. Savchuk Ivano-Frankivsk National Technical University of Oil and Gas
  • A.I. Bandura Ivano-Frankivsk National Tecnical University of OIl and Gas
Keywords: entire curve;, Picard exceptional vector;, asymptotic vector;, meromorphic function;, asymptotic value;, Picard exceptional value

Abstract

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate
a relationship between the asymptotic vectors and the Picard exceptional vectors.

A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that
$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$.

We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.
Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph
``Value Distribution of Meromorphic Functions''.
Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.

At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

Author Biography

A.I. Bandura, Ivano-Frankivsk National Tecnical University of OIl and Gas

Ivano-Frankivsk National Tecnical University of OIl and Gas

References

F. Iversen, Recherches sur les fonctions inverses des fonctions meromorphes, These, Helsingfors, 1914, 1–67.

L. Ahlfors, ¨ Uber die asymptotischen Werte der meromorphen Funktionen endlicher Ordnung, Acta Acad. Aboensis. Math. et Phys., 6 (1932), №9, 1–8.

A.A. Gol’dberg, I V. Ostrovskii, Value Distribution of Meromorphic Functions, Providence: AMS, 2008. (Translated from Russian ed. Moscow, Nauka, 1970).

V.P. Petrenko, Entire curves, Kharkiv: Vyshcha shkola, 1984 (in Russian).

Ya.I. Savchuk, Structure of the set of defect vectors of entire and analytic curves of finite order, Ukr. Math. J., 37 (1985), №5, 494–499. doi: 10.1007/BF01061174

A.I. Bandura, Ya.I. Savchuk, Structure of the set of Borel exceptional vectors for entire curves, Mat. Stud., 53 (2020), №1, 41–47. doi: 10.30970/ms.53.1.41-47

K. Boussaf, Picard values of p-adic meromorphic functions, P-Adic Num. Ultrametr. Anal. Appl., 2 (2010), 285–292. doi: 10.1134/S2070046610040035

I.M. Dektyarev, Averaged deficiencies of holomorphic curves and divisors with an excessive deficiency value, Russian Mathematical Surveys, 44 (1989), №1, 237–238. doi: 10.1070/RM1989v044n01 ABEH002015

Mori S., Topics on meromorphic mappings and defects, Complex Var. Elliptic Equ., 56 (2011), №1-4, 363–373. doi: 10.1080/17476930903394903

Ya.I. Savchuk, Set of deficient vectors of integral curves, Ukr. Math. J., 35 (1983), №3, 334–338. doi: 10.1007/BF01092190

Ya.I. Savchuk, Inverse problem of the theory of distribution of the values of entire and analytic curves, J. Soviet Mathematics, 48 (1990), №2, 220–231. doi: 10.1007/BF01095801

Ya.I. Savchuk, Valiron deficient vectors of entire curves of finite order, J. Soviet Mathematics, 52 (1990), №5, 3435–3437. doi: 10.1007/BF01099913

N. Toda, Holomorphic curves with an infinite number of deficiencies, Proc. Japan Acad. Ser. A Math. Sci., 80 (2004), №6, 90–95. doi: 10.3792/pjaa.80.90.

Published
2021-10-23
How to Cite
1.
Savchuk Y, Bandura A. Asymptotic vectors of entire curves. Mat. Stud. [Internet]. 2021Oct.23 [cited 2022Jan.20];56(1):48-4. Available from: http://www.matstud.org.ua/ojs/index.php/matstud/article/view/254
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Articles