• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.137-147
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• 2015

Let f be a transcendental meromorphic function of finite order in the plane such that $f^{(m)}$ has finitely many zeros for some positive integer $m{\geq}2$. Suppose that $f^{(k)}$ and f share a CM, where $k{\geq}1$ is a positive integer, $a{\neq}0$ is a finite complex value. Then f is an entire function such that $f^{(k)}-a=c(f-a)$, where $c{\neq}0$ is a nonzero constant. The results in this paper are concerning a conjecture of Bruck [5]. An example is provided to show that the results in this paper, in a sense, are the best possible.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.307-318
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• 2018

In this paper, we obtain $Fekete-Szeg{\ddot{o}}$ inequalities for certain class of meromorphic functions f(z) for which $-{\frac{(1-{\frac{{\alpha}}{q}})qzD_qf(z)+{\alpha}qzD_q[zD_qf(z)]}{(1-{\frac{{\alpha}}{q}})f(z)+{\alpha}zD_qf(z)}{\prec}{\varphi}(z)$(${\alpha}{\in}{\mathbb{C}}{\backslash}(0,1]$, 0 < q < 1). Sharp bounds for the $Fekete-Szeg{\ddot{o}}$ functional ${\mid}{\alpha}_1-{\mu}{\alpha}^2_0{\mid}$ are obtained.

• Kyungpook Mathematical Journal
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• v.52 no.1
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• pp.39-47
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• 2012

Let $\mathcal{F}$ be a family of meromorphic functions in a domain D, and let $k$, $n({\geq}2)$ be two positive integers, and let $S=\{a_1,a_2,{\ldots},a_n\}$, where $a_1$, $a_2$, ${\ldots}$, $a_n$ are distinct finite complex numbers. If for each $f{\in}\mathcal{F}$, all zeros of $f$ have multiplicity at least $k+1$, $f$ and $G(f)$ share the set $S$ in $D$, where $G(f)=P(f^{(k)})+H(f)$ is a differential polynomial of $f$, then$\mathcal{F}$ is normal in $D$.

• Journal of the Korean Mathematical Society
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• v.35 no.1
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• pp.127-134
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• 1998

It is proved [6] that there exists a basis of $L^\Gamma$ (the space of meromorphic vector fields on a Riemann surface, holomorphic away from two fixed points) represented by the vector fields which have the expected zero or pole order at the two points. In this paper, we carry out the same task for the quadratic differentials. More precisely, we compute a basis of $Q^\Gamma$ (the sapce of meromorphic quadratic differentials on a Riemann surface, holomorphic away from two fixed points). This basis consists of the quadratic differentials which have the expected zero or pole order at the two points. Furthermore, we show that $Q^\Gamma$ has a Lie algebra structure which is induced from the Krichever-Novikov algebra $L^\Gamma$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.6
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• pp.1541-1565
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• 2020

Let f be a meromorphic function of finite order ρ with few poles in the sense S_λ(r, f) := O(r^λ+ε) + S(r, f), where λ < ρ and ε ∈ (0, ρ - λ), and let g(f) := Σ^k_j=1b_j(z)f^(k_j)(z + c_j) be a linear delay-differential polynomial of f with small meromorphic coefficients b_j in the sense S_λ(r, f). The zero distribution of fⁿ(g(f))^s - b₀ is considered in this paper, where b₀ is a small function in the sense S_λ(r, f).

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1175-1192
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• 2021

In this paper, we have studied on the uniqueness problems of meromorphic functions with its linear c-shift operator in the light of partial sharing. Our two results improve and generalize two very recent results of Noulorvang-Pham [Bull. Korean Math. Soc. 57 (2020), no. 5, 1083-1094] in some sense. In addition, our other results have improved and generalized a series of results due to Lü-Lü [Comput. Methods Funct. Theo. 17 (2017), no. 3, 395-403], Zhen [J. Contemp. Math. Anal. 54 (2019), no. 5, 296-301] and Banerjee-Bhattacharyya [Adv. Differ. Equ. 509 (2019), 1-23]. We have exhibited a number of examples to show that some conditions used in our results are essential.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.439-455
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• 2019

Let ${\Sigma}_k(p)$ be the class of univalent meromorphic functions defined on the unit disc ${\mathbb{D}}$ with k-quasiconformal extension to the extended complex plane ${\hat{\mathbb{C}}}$, where $0{\leq}k<1$. Let ${\Sigma}^0_k(p)$ be the class of functions $f{\in}{\Sigma}_k(p)$ having expansion of the form $f(z)=1/(z-p)+{\sum_{n=1}^{\infty}}\;b_nz^n$ on ${\mathbb{D}}$. In this article, we obtain sharp area distortion and weighted area distortion inequalities for functions in ${\sum_{k}^{0}}(p)$. As a consequence of the obtained results, we present a sharp upper bound for the Hilbert transform of characteristic function of a Lebesgue measurable subset of ${\mathbb{D}}$.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.45-56
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• 2019

The aim of this paper is to investigate the problems of meromorphic functions sharing values concerning periodicities and shifts. In this paper we prove the following result: Let f(z) and g(z) be two nonconstant entire functions, let $c{\in}{\mathbb{C}}{\setminus}\{0\}$, and let $a_1$, $a_2$ be two distinct finite complex numbers. Suppose that ${\mu}(f){\neq}1$, ${\rho}_2(f)<1$, and f(z) = f(z+c) for all $z{\in}{\mathbb{C}}$. If f(z) and g(z) share $a_1$ CM, $a_2$ IM, then $f(z){\equiv}g(z)$. Moreover, examples are given to show that all the conditions are necessary.

• 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)e^α(z) and fⁿ + P_*(f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(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]).

• The Pure and Applied Mathematics
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• v.28 no.4
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• pp.355-376
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• 2021

In this paper we wish to establish the integral representations of generalized relative order (𝛼, 𝛽) and generalized relative type (𝛼, 𝛽) of entire and meromorphic functions where 𝛼 and 𝛽 are continuous non-negative functions defined on (-∞, +∞). We also investigate their equivalence relation under some certain condition.