• Kyungpook Mathematical Journal
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• 제51권2호
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• pp.145-164
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• 2011

In the paper, we study with weighted sharing method the uniqueness of entire functions concerning nonlinear differential polynomials sharing one value and prove two uniqueness theorems, first one of which generalizes some recent results in [10] and [16]. Our second theorem will supplement a result in [17].

• 호남수학학술지
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• 제38권1호
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• pp.169-212
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• 2016

Entire functions have been investigated so popularly to have been divided into a large number of specialized subjects. Even the limited subject of entire functions is too vast to be dealt with in a single volume with any approach to completeness. Here, in this paper, we choose to investigate certain interesting results associated with the comparative growth properties of iterated entire functions using (p, q)-th order and (p, q)-th lower order, in a rather comprehensive and systematic manner.

• 대한수학회보
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• 제58권4호
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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]).

• Kyungpook Mathematical Journal
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• 제55권1호
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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.

• East Asian mathematical journal
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• 제24권4호
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• pp.399-405
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• 2008

In this paper, we get another proof of L. A. Rubel and C. C. Yang's theorem. It is more vivid technique. By using the relation between normal families and shared values we prove the theorem.

• 대한수학회논문집
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• 제39권1호
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• pp.117-125
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• 2024

In this paper, our focus is on exploring value sharing problems related to a transcendental entire function f and its associated differential-difference polynomials. We aim to establish some results which are related to differential-difference counterpart of the Brück conjecture.

• 대한수학회보
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• 제48권5호
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• pp.951-957
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• 2011

In 1996, Br$\ddot{u}$ck studied the relation between f and f' if an entire function f shares one value a CM with its first derivative f' and posed the famous Br$\ddot{u}$ck conjecture. In this work, we generalize the value a in the Br$\ddot{u}$ck conjecture to a small function ${\alpha}$. Meanwhile, we prove that the Br$\ddot{u}$ck conjecture holds for a class of meromorphic functions.

• 대한수학회논문집
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• 제35권1호
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• pp.229-241
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• 2020

For a second order linear differential equation f" + A(z)f' + B(z)f = 0, with A(z) and B(z) being transcendental entire functions under some restrictions, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions, with a condition, have exponent of convergence of zeros equal to infinity. Also, we have extended these results to higher order linear differential equations.

• 대한수학회보
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• 제56권1호
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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.

• 대한수학회보
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• 제53권2호
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• pp.411-421
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• 2016

In this paper, we investigate the problem of transcendental entire functions that share two values with one of their derivative. Let f be a transcendental entire function, n and k be two positive integers. If $f^n-Q_1$ and $(f^n)^{(k)}-Q_2$ share 0 CM, and $n{\geq}k+1$, then $(f^n)^{(k)}{\equiv}{\frac{Q_2}{Q_1}}f^n$. Furthermore, if $Q_1=Q_2$, then $f=ce^{\frac{\lambda}{n}z}$, where $Q_1$, $Q_2$ are polynomials with $Q_1Q_2{\not\equiv}0$, and c, ${\lambda}$ are non-zero constants such that ${\lambda}^k=1$. This result shows that the Conjecture given by W. $L{\ddot{u}}$, Q. Li and C. Yang [On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc. 51 (2014), no. 5, 1281-1289.] is true. Also we exhibit some examples to show that the conditions of our result are the best possible.