• Title/Summary/Keyword: transcendental entire function

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Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

  • Sahoo, Pulak;Biswas, Gurudas
    • Kyungpook Mathematical Journal
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    • v.58 no.3
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    • pp.519-531
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    • 2018
  • In this paper, we investigate the uniqueness problem of entire functions sharing two polynomials with their k-th derivatives. We look into the conjecture given by $L{\ddot{u}}$, Li and Yang [Bull. Korean Math. Soc., 51(2014), 1281-1289] for the case $F=f^nP(f)$, where f is a transcendental entire function and $P(z)=a_mz^m+a_{m-1}z^{m-1}+{\ldots}+a_1z+a_0({\not{\equiv}}0)$, m is a nonnegative integer, $a_m,a_{m-1},{\ldots},a_1,a_0$ are complex constants and obtain a result which improves and generalizes many previous results. We also provide some examples to show that the conditions taken in our result are best possible.

ON THE SHAPE OF MAXIMUM CURVE OF eaz2+bz+c

  • KIM, MIHWA;KIM, JEONG-HEON
    • Journal of applied mathematics & informatics
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    • v.35 no.1_2
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    • pp.75-82
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    • 2017
  • In this paper, we investigate the proper shape and location of the maximum curve of transcendental entire functions $e^{az^2+bz+c}$. We show that the alpha curve of $e^{az^2+bz+c}$ is a subset of a rectangular hyperbola, and the maximum curve is the connected set originating from the origin as a subset of the alpha curve.

THREE RESULTS ON TRANSCENDENTAL MEROMORPHIC SOLUTIONS OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

  • Li, Nan;Yang, Lianzhong
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.795-814
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    • 2021
  • In this paper, we study the transcendental meromorphic solutions for the nonlinear differential equations: fn + P(f) = R(z)eα(z) and fn + P*(f) = p1(z)eα1(z) + p2(z)eα2(z) in the complex plane, where P(f) and P*(f) are differential polynomials in f of degree n - 1 with coefficients being small functions and rational functions respectively, R is a non-vanishing small function of f, α is a nonconstant entire function, p1, p2 are non-vanishing rational functions, and α1, α2 are nonconstant polynomials. Particularly, we consider the solutions of the second equation when p1, p2 are nonzero constants, and deg α1 = deg α2 = 1. Our results are improvements and complements of Liao ([9]), and Rong-Xu ([11]), etc., which partially answer a question proposed by Li ([7]).

SHARED VALUES AND BOREL EXCEPTIONAL VALUES FOR HIGH ORDER DIFFERENCE OPERATORS

  • Liao, Liangwen;Zhang, Jie
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.1
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    • pp.49-60
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    • 2016
  • In this paper, we investigate the high order difference counterpart of $Br{\ddot{u}}ck^{\prime}s$ conjecture, and we prove one result that for a transcendental entire function f of finite order, which has a Borel exceptional function a whose order is less than one, if ${\Delta}^nf$ and f share one small function d other than a CM, then f must be form of $f(z)=a+ce^{{\beta}z}$, where c and ${\beta}$ are two nonzero constants such that $\frac{d-{\Delta}^na}{d-a}=(e^{\beta}-1)^n$. This result extends Chen's result from the case of ${\sigma}(d)$ < 1 to the general case of ${\sigma}(d)$ < ${\sigma}(f)$.

DYNAMICAL PROPERTIES ON THE ITERATION OF CF-FUNCTIONS

  • Yoo, Seung-Jae
    • Journal of the Chungcheong Mathematical Society
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    • v.12 no.1
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    • pp.1-13
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    • 1999
  • The purpose of this paper is to show that if the Fatou set F(f) of a CF-meromorphic function f has two completely invariant components, then they are the only components of F(f) and that the Julia set of the entire transcendental function $E_{\lambda}(z)={\lambda}e^z$ for 0 < ${\lambda}$ < $\frac{1}{e}$ contains a Cantor bouquet by employing the Devaney and Tangerman's theorem[10].

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Growth order of Meromorphic Solutions of Higher-order Linear Differential Equations

  • Xu, Junfeng;Zhang, Zhanliang
    • Kyungpook Mathematical Journal
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    • v.48 no.1
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    • pp.123-132
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    • 2008
  • In this paper, we investigate higher-order linear differential equations with entire coefficients of iterated order. We improve and extend the result of L. Z. Yang by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen and the extended Wiman-Valiron theory by Wang and Yi. We also consider the nonhomogeneous linear differential equations.

ON THE UNIQUENESS OF CERTAIN TYPE OF SHIFT POLYNOMIALS SHARING A SMALL FUNCTION

  • Saha, Biswajit
    • Korean Journal of Mathematics
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    • v.28 no.4
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    • pp.889-906
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    • 2020
  • In this article, we consider the uniqueness problem of the shift polynomials $f^n(z)(f^m(z)-1){\prod\limits_{j=1}^{s}}f(z+c_j)^{{\mu}_j}$ and $f^n(z)(f(z)-1)^m{\prod\limits_{j=1}^{s}}f(z+c_j)^{{\mu}_j}$, where f(z) is a transcendental entire function of finite order, cj (j = 1, 2, …, s) are distinct finite complex numbers and n(≥ 1), m(≥ 1), s and µj (j = 1, 2, …, s) are integers. With the concept of weakly weighted sharing and relaxed weighted sharing we obtain some results which extend and generalize some results due to P. Sahoo [Commun. Math. Stat. 3 (2015), 227-238].

RADIAL OSCILLATION OF LINEAR DIFFERENTIAL EQUATION

  • Wu, Zhaojun
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.5
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    • pp.911-921
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    • 2012
  • In this paper, the radial oscillation of the solutions of higher order homogeneous linear differential equation $$f^{(k)}+A_{n-2}(z)f^{(k-2)}+{\cdots}+A_1(z)f^{\prime}+A_0(z)f=0$$ with transcendental entire function coefficients is studied. Results are obtained to extend some results in [Z. Wu and D. Sun, Angular distribution of solutions of higher order linear differential equations, J. Korean Math. Soc. 44 (2007), no. 6, 1329-1338].

SOME RESULTS ON COMPLEX DIFFERENTIAL-DIFFERENCE ANALOGUE OF BRÜCK CONJECTURE

  • Chen, Min Feng;Gao, Zong Sheng
    • Communications of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.361-373
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    • 2017
  • In this paper, we utilize the Nevanlinna theory and uniqueness theory of meromorphic function to investigate the differential-difference analogue of $Br{\ddot{u}}ck$ conjecture. In other words, we consider ${\Delta}_{\eta}f(z)=f(z+{\eta})-f(z)$ and f'(z) share one value or one small function, and then obtain the precise expression of transcendental entire function f(z) under certain conditions, where ${\eta}{\in}{\mathbb{C}}{\backslash}\{0\}$ is a constant such that $f(z+{\eta})-f(z){\not\equiv}0$.