• Title/Summary/Keyword: small functions

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TWO MEROMORPHIC FUNCTIONS SHARING FOUR PAIRS OF SMALL FUNCTIONS

  • Nguyen, Van An;Si, Duc Quang
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1159-1171
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    • 2017
  • The purpose of this paper is twofold. The first is to show that two meromorphic functions f and g must be linked by a quasi-$M{\ddot{o}}bius$ transformation if they share a pair of small functions regardless of multiplicity and share other three pairs of small functions with multiplicities truncated to level 4. We also show a quasi-$M{\ddot{o}}bius$ transformation between two meromorphic functions if they share four pairs of small functions with multiplicities truncated by 4, where all zeros with multiplicities at least k > 865 are omitted. Moreover the explicit $M{\ddot{o}}bius$-transformation between such f and g is given. Our results are improvement of some recent results.

ON THE MULTIPLE VALUES AND UNIQUENESS OF MEROMORPHIC FUNCTIONS SHARING SMALL FUNCTIONS AS TARGETS

  • Cao, Ting-Bin;Yi, Hong-Xun
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.631-640
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    • 2007
  • The purpose of this article is to deal with the multiple values and uniqueness of meromorphic functions with small functions in the whole complex plane. We obtain a more general theorem which improves and extends strongly the results of R. Nevanlinna, Li-Qiao, Yao, Yi, and Thai-Tan.

UNICITY OF MERMORPHIC FUNCTIONS CONCERNING SHARED FUNCTIONS WITH THEIR DIFFERENCE

  • Deng, Bingmao;Fang, Mingliang;Liu, Dan
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1511-1524
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    • 2019
  • In this paper, we investigate the uniqueness of meromorphic functions of finite order concerning sharing small functions and prove that if f(z) and ${\Delta}_cf(z)$ share a(z), b(z), ${\infty}$ CM, where a(z), b(z)(${\neq}{\infty}$) are two distinct small functions of f(z), then $f(z){\equiv}{\Delta}_cf(z)$. The result improves the results due to Li et al. ([9]), Cui et al. ([1]) and $L{\ddot{u}}$ et al. ([12]).

SOME RESULTS ON MEROMORPHIC SOLUTIONS OF CERTAIN NONLINEAR DIFFERENTIAL EQUATIONS

  • Li, Nan;Yang, Lianzhong
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1095-1113
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    • 2020
  • In this paper, we investigate the transcendental meromorphic solutions for the nonlinear differential equations $f^nf^{(k)}+Q_{d_*}(z,f)=R(z)e^{{\alpha}(z)}$ and fnf(k) + Qd(z, f) = p1(z)eα1(z) + p2(z)eα2(z), where $Q_{d_*}(z,f)$ and Qd(z, f) are differential polynomials in f with small functions as coefficients, of degree d* (≤ n - 1) and d (≤ n - 2) respectively, R, p1, p2 are non-vanishing small functions of f, and α, α1, α2 are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of these kinds of meromorphic solutions and their possible forms of the above equations.

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]).