• 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 $f(z+{\eta})-f(z){\not\equiv}0$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.559-581
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• 2013

This paper is motivated by a 2001 paper of Choie and Kim and a 2006 paper of Bump and Choie. The paper of Choie and Kim extends an earlier result of Bol for elliptic modular forms to the setting of Siegel and Jacobi forms. The paper of Bump and Choie provides a representation theoretic interpretation of the phenomenon, and shows how a natural generalization of Choie and Kim's result on Siegel modular forms follows from a natural conjecture regarding ($g$, K)-modules. In this paper, it is shown that the conjecture of Bump and Choie follows from work of Boe. A second proof which is along the lines of the proof given by Bump and Choie in the genus 2 case is also included, as is a similar treatment of the result of Choie and Kim on Jacobi forms.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.439-446
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• 2015

In this paper we consider the meromorphic function G(z) with a pole of order 1 at -a and analytic function F(z) with a zero -a of order 2 in $\mathbb{D}=\{z :{\mid}z{\mid}<1\}$, where -1 < a < 1. From these functions we obtain the regular simply-connected minimal surface $S=\{(u(z),\;{\nu}(z),\;H(z)):z{\in}\mathbb{D}\}$ in $E^3$ and the harmonic function $f=u+i{\nu}$ defined on $\mathbb{D}$, and then we investigate properties of the minimal surface S and the harmonic function f.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• Bulletin of the Korean Mathematical Society
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• v.55 no.5
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• pp.1529-1561
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• 2018

Let f and g be nonconstant meromorphic (entire, respectively) functions in the complex plane such that f and g are of finite order, let a and b be nonzero complex numbers and let n be a positive integer satisfying $n{\geq}21$ ($n{\geq}12$, respectively). We show that if the difference polynomials $f^n(z)+af(z+{\eta})$ and $g^n(z)+ag(z+{\eta})$ share b CM, and if f and g share 0 and ${\infty}$ CM, where ${\eta}{\neq}0$ is a complex number, then f and g are either equal or at least closely related. The results in this paper are difference analogues of the corresponding results from.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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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.

• East Asian mathematical journal
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• v.37 no.1
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• pp.87-95
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• 2021

Let K be an imaginary quadratic field with ��K its ring of integers. For a positive integer N, let K(N) be the ray class field of K modulo N��K, and let ��N be the field of meromorphic modular functions of level N whose Fourier coefficients lie in the Nth cyclotomic field. For each h ∈ ��N, we construct a ray class invariant as its special value in terms of the extended form class group, and show that the invariant satisfies the natural transformation formula via the Artin map in the sense of Siegel and Stark. Finally, we establish an isomorphism between the extended form class group and Gal(K(N)/K) without any restriction on K.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.155-166
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• 2022

In this paper, the paired Hayman conjecture of different types are considered, namely, the zeros distribution of f(z)ⁿL(g) - a(z) and g(z)ⁿL(f) - a(z), where L(h) takes the derivatives h^(k)(z) or the shift h(z+c) or the difference h(z+c)-h(z) or the delay-differential h^(k)(z+c), where k is a positive integer, c is a non-zero constant and a(z) is a nonzero small function with respect to f(z) and g(z). The related uniqueness problems of complex delay-differential polynomials are also considered.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.659-675
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• 2023

Let K be an algebraically closed field of characteristic 0 and let f be a non-fibered planar quadratic polynomial map of topological degree 2 defined over K. We assume further that the meromorphic extension of f on the projective plane has the unique indeterminacy point. We define the critical pod of f where f sends a critical point to another critical point. By observing the behavior of f at the critical pod, we can determine a good conjugate of f which shows its statue in GIT sense.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1651-1672
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• 2023

In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u₀, u₁, . . . , u_m, v₀, v₁, v₂) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u₀, u₁, . . ., u_n, v) with coefficients in a set L_δ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ^(m)(z), Γ(z), Γ'(z), . . . , Γ⁽ⁿ⁾(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is any non-trivial polynomial in the m + n + 2 variables u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u₀, u₁, . . . , u_m, v₀, v₁, . . . , v_n) is a distinguished polynomial in the n + 1 variables v₀, v₁, . . . , v_n. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.