• Title/Summary/Keyword: transcendental entire function

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THE ITERATION OF ENTIRE FUNCTION

  • Sun, Jianwu
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.369-378
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    • 2001
  • In this paper, we obtain the following results: Let f be a transcendental entire function with log M(r,f)=$O(log r)^\beta (e^{log r}^\alpha)\; (0\leq\alpha<1,\beta>1$). Then every component of N(f) is bounded. This result generalizes the result of Baker.

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ON THE TRANSCENDENTAL ENTIRE SOLUTIONS OF A CLASS OF DIFFERENTIAL EQUATIONS

  • Lu, Weiran;Li, Qiuying;Yang, Chungchun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1281-1289
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    • 2014
  • In this paper, we consider the differential equation $$F^{\prime}-Q_1=Re^{\alpha}(F-Q_2)$$, where $Q_1$ and $Q_2$ are polynomials with $Q_1Q_2{\neq}0$, R is a rational function and ${\alpha}$ is an entire function. We consider solutions of the form $F=f^n$, where f is an entire function and $n{\geq}2$ is an integer, and we prove that if f is a transcendental entire function, then $\frac{Q_1}{Q_2}$ is a polynomial and $f^{\prime}=\frac{Q_1}{nQ_2}f$. This theorem improves some known results and answers an open question raised in [16].

UNIQUENESS RELATED TO HIGHER ORDER DIFFERENCE OPERATORS OF ENTIRE FUNCTIONS

  • Xinmei Liu;Junfan Chen
    • The Pure and Applied Mathematics
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    • v.30 no.1
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    • pp.43-65
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    • 2023
  • In this paper, by using the difference analogue of Nevanlinna's theory, the authors study the shared-value problem concerning two higher order difference operators of a transcendental entire function with finite order. The following conclusion is proved: Let f(z) be a finite order transcendental entire function such that λ(f - a(z)) < ρ(f), where a(z)(∈ S(f)) is an entire function and satisfies ρ(a(z)) < 1, and let 𝜂(∈ ℂ) be a constant such that ∆𝜂n+1 f(z) ≢ 0. If ∆𝜂n+1 f(z) and ∆𝜂n f(z) share ∆𝜂n a(z) CM, where ∆𝜂n a(z) ∈ S ∆𝜂n+1 f(z), then f(z) has a specific expression f(z) = a(z) + BeAz, where A and B are two non-zero constants and a(z) reduces to a constant.

DYNAMICS OF TRANSCENDENTAL ENTIRE FUNCTIONS WITH SIEGEL DISKS AND ITS APPLICATIONS

  • Katagata, Koh
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.4
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    • pp.713-724
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    • 2011
  • We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

On the Value Distribution of ff(k)

  • Wang, Jian-Ping
    • Kyungpook Mathematical Journal
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    • v.46 no.2
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    • pp.169-180
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    • 2006
  • This paper proves the following results: Let $f$ be a transcendental entire function, and let $k({\geq})2$ be a positive integer. If $T(r,\;f){\neq}N_{1)}(r,1/f)+S(r,\;f)$, then $ff^{(k)}$ assumes every finite nonzero value infinitely often. Also the case when f is a transcendental meromorphic function has been considered and some results are obtained.

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A RESULT ON A CONJECTURE OF W. LÜ, Q. LI AND C. YANG

  • Majumder, Sujoy
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.411-421
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    • 2016
  • In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.

SINGULARITIES AND STRICTLY WANDERING DOMAINS OF TRANSCENDENTAL SEMIGROUPS

  • Huang, Zhi Gang;Cheng, Tao
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.343-351
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    • 2013
  • In this paper, the dynamics on a transcendental entire semigroup G is investigated. We show the possible values of any limit function of G in strictly wandering domains and Fatou components, respectively. Moreover, if G is of class $\mathfrak{B}$, for any $z$ in a Fatou domain, there does not exist a sequence $\{g_k\}$ of G such that $g_k(z){\rightarrow}{\infty}$ as $k{\rightarrow}{\infty}$.

ON THE NORMALITY OF TRANSLATED FAMILIES OF TRANSCENDENTAL ENTIRE FUNCTIONS

  • KIM JEONG HEON;KWON KI HO;PARK SUK BONG
    • Journal of applied mathematics & informatics
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    • v.17 no.1_2_3
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    • pp.573-583
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    • 2005
  • For a certain set G in the complex plane, we construct a transcendental entire function f whose translated family ${f(2^{n}z)}$ is normal only at z in G and establish the relation between the normal family and the Julia direction of f(z).