• 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) + Q_d(z, f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z), where $Q_{d_*}(z,f)$ and Q_d(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.

• Bulletin of the Korean Mathematical Society
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• v.51 no.6
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• pp.1735-1748
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• 2014

In this paper, we investigate the finite order transcendental meromorphic solutions of complex difference equation of Malmquist type $$\prod_{i=1}^{n}f(z+c_i)=R(z,f)$$, where $c_1,{\ldots},c_n{\in}\mathbb{C}{\backslash}\{0\}$, and R(z, f) is an irreducible rational function in f(z) with meromorphic coefficients. We obtain some results on deficiencies of the solutions. Using these results, we prove that the growth order of the finite order solution f(z) is 1, if f(z) has Borel exceptional values $a({\in}\mathbb{C})$ and ${\infty}$. Moreover, we give the forms of f(z).

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.593-605
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• 2017

In this paper, we consider some normality criteria concerning multiple values. Let $\mathcal{F}$ be a family of meromorphic functions defined in a domain D. Let k be a positive integer and ${\psi}(z){\not\equiv}0$, ${\infty}$ be a meromorphic function in D. If, for each $f{\in}\mathcal{F}$ and $z{\in}D$, (1) $f(z){\neq}0$, and all of whose poles are multiple; (2) all zeros of $f^{(k)}(z)-{\psi}(z)$ have multiplicities at least k + 3 in D; (3) all poles of ${\psi}(z)$ have multiplicities at most k in D, then $\mathcal{F}$ is normal in D.

• Bulletin of the Korean Mathematical Society
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• v.50 no.3
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• pp.731-745
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• 2013

In this paper, we investigate uniqueness problems of certain types of $q$-difference polynomials, which improve some results in [20]. However, our proof is different from that in [20]. Moreover, we obtain a uniqueness result in the case where $q$-differences of two entire functions share values as well. This research also shows that there exist two sets, such that for a zero-order non-constant meromorphic function $f$ and a non-zero complex constant $q$, $E(S_j,f)=E(S_j,{\Delta}_qf)$ for $j=1,2$ imply $f(z)=t{\Delta}_qf$, where $t^n=1$. This gives a partial answer to a question of Gross concerning a zero order meromorphic function $f(z)$ and $t{\Delta}_qf$.

• 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.

• 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.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.959-970
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• 2017

In this paper, we investigate the transcendental meromorphic solutions with finite order of two different types of difference equations $${\sum\limits_{j=1}^{n}}a_jf(z+c_j)={\frac{P(z,f)}{Q(z,f)}}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ and $${\prod\limits_{j=1}^{n}}f(z+c_j)={\frac{P(z,f)}{Q(z,f)}={\frac{{\sum_{k=0}^{p}}b_kf^k}{{\sum_{l=0}^{q}}d_lf^l}}$$ that share three distinct values with another meromorphic function. Here $a_j$, $b_k$, $d_l$ are small functions of f and $a_j{\not{\equiv}}(j=1,2,{\ldots},n)$, $b_p{\not{\equiv}}0$, $d_q{\not{\equiv}}0$. $c_j{\neq}0$ are pairwise distinct constants. p, q, n are non-negative integers. P(z, f) and Q(z, f) are two mutually prime polynomials in f.

• Kyungpook Mathematical Journal
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• v.55 no.2
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• pp.411-427
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• 2015

In this article, we main study the uniqueness problem of meromorphic function which difference polynomials sharing common values. We consider the entire function $(f^n(f^m-1)\prod_{j=1}^{s}f(z+c_j)^{{\mu}j})^{(k)}$ and the meromorphic function $f^n(f^m-1)\prod_{j=1}^{s}f(z+c_j)^{{\mu}j}$ to get the main results which extend Theorem 1.1 in paper[5] and theorem 1.4 in paper[6].

• Kyungpook Mathematical Journal
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• v.55 no.3
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• pp.641-652
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• 2015

In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain $D{\subseteq}{\mathbb{C}}$ and n, k be two positive integers such that $n{\geq}k+1$, and let a, b be two finite complex constants such that $a{\neq}0$. Suppose that (1) $f+a(f^{(k)})^n$ and $g+a(g^{(k)})^n$ share b in D for every pair of functions f, $g{\in}F$; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each $f{\in}F$ in D; (3) Zeros of $f^{(k)}(z)$ are not the b points of f(z) for each $f{\in}F$ in D. Then F is normal in D. And some examples are provided to show the result is sharp.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.235-243
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• 2009

In this paper, we deal with the problem of uniqueness of meromorphic functions that share two small functions with their derivatives, and obtain the following result which improves a result of Yao and Li: Let f(z) be a nonconstant meromorphic function, k > 5 be an integer. If f(z) and g(z) = $a_1(z)f(z)+a_2(z)f^{(k)}(z)$ share the value 0 CM, and share b(z) IM, $\overline{N}_E(r,f=0=F^{(k)})=S(r)$, f${\equiv}$g, where $a_1(z)$, $a_2(z)$ and b(z) are small functions of f(z).