• Title, Summary, Keyword: Nevanlinna theory

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ON SOME SPECIAL DIFFERENCE EQUATIONS OF MALMQUIST TYPE

  • Zhang, Jie
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.51-61
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    • 2018
  • In this article, we mainly use Nevanlinna theory to investigate some special difference equations of malmquist type such as $f^2+({\Delta}_cf)^2={\beta}^2$, $f^2+({\Delta}_cf)^2=R$, $f{^{\prime}^2}+({\Delta}_cf)^2=R$ and $f^2+(f(z+c))^2=R$, where ${\beta}$ is a nonzero small function of f and R is a nonzero rational function respectively. These discussions extend one related result due to C. C. Yang et al. in some sense

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 $f(z+{\eta})-f(z){\not\equiv}0$.

SECOND MAIN THEOREM AND UNIQUENESS PROBLEM OF ZERO-ORDER MEROMORPHIC MAPPINGS FOR HYPERPLANES IN SUBGENERAL POSITION

  • Luong, Thi Tuyet;Nguyen, Dang Tuyen;Pham, Duc Thoan
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.205-226
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    • 2018
  • In this paper, we show the Second Main Theorems for zero-order meromorphic mapping of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n({\mathbb{C}})$ intersecting hyperplanes in subgeneral position without truncated multiplicity by considering the p-Casorati determinant with $p{\in}{\mathbb{C}}^m$ instead of its Wronskian determinant. As an application, we give some unicity theorems for meromorphic mapping under the growth condition "order=0". The results obtained include p-shift analogues of the Second Main Theorem of Nevanlinna theory and Picard's theorem.

DIFFERENTIAL EQUATIONS RELATED TO FAMILY A

  • Li, Ping;Meng, Yong
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.2
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    • pp.247-260
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    • 2011
  • Let h be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in h with the coefficients being small functions of h. We prove that if f is a meromorphic function and if $f^m$ is identically a polynomial in h with the constant term not vanish identically, then f is a polynomial in h. As an application, we are able to find the entire solutions of the differential equation of the type $$f^n+P(f)=be^{sz}+Q(e^z)$$, where P(f) is a differential polynomial in f of degree at most n-1, and Q($e^z$) is a polynomial in $e^z$ of degree k $\leqslant$ max {n-1, s(n-1)/n} with small functions of $e^z$ as its coefficients.

EXPRESSIONS OF MEROMORPHIC SOLUTIONS OF A CERTAIN TYPE OF NONLINEAR COMPLEX DIFFERENTIAL EQUATIONS

  • Chen, Jun-Fan;Lian, Gui
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.1061-1073
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    • 2020
  • In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^n+Qd(z,f)=\sum\limits_{i=1}^{3}pi(z)e^{{\alpha}_i(z)}$$ are studied by using Nevanlinna theory, where n ≥ 5 is an integer, Qd(z, f) is a differential polynomial in f of degree d ≤ n - 4 with rational functions as its coefficients, p1(z), p2(z), p3(z) are non-vanishing rational functions, and α1(z), α2(z), α3(z) are nonconstant polynomials such that α'1(z), α'2(z), α'3(z) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

Entire Functions That Share One Value With Their Derivatives

  • Lu, Feng;Xu, Junfeng
    • Kyungpook Mathematical Journal
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    • v.47 no.3
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    • pp.439-448
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    • 2007
  • In the paper, we use the theory of normal family to study the problem on entire function that share a finite non-zero value with their derivatives and prove a uniqueness theorem which improve the result of J.P. Wang and H.X. Yi.

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