• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.311-322
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• 2006

For the Briot-Bouquet differential equations of the form given in [1] $${{\mu}(z)+\frac {z{\mu}'(z)}{z\frac {f'(z)}{f(z)}\[\alpha{\mu}(z)+\beta]}=g(z)$$ we can reduce them to $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$ where $$v(z)=\alpha{\mu}(z)+\beta,\;h(z)={\alpha}g(z)+\beta\;and\;F(z)=f(z)/f'(z)$$. In this paper we are going to give conditions in order that if u and v satisfy, respectively, the equations (1) $${{\mu}(z)+F(z)\frac {v'(z)}{v(z)}=h(z)$$, $${{\mu}(z)+G(z)\frac {v'(z)}{v(z)}=g(z)$$ with certain conditions on the functions F and G applying the concept of strong subordination $g\;\prec\;\prec\;h$ given in [2] by the author, implies that $v\;\prec\;{\mu},\;where\;\prec$ indicates subordination.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1235-1255
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• 2019

In this paper, we introduce SR-additive codes as a generalization of the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes, where S is an R-algebra and an SR-additive code is an R-submodule of $S^{\alpha}{\times}R^{\beta}$. In particular, the definitions of bilinear forms, weight functions and Gray maps on the classes of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$ and ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-additive codes are generalized to SR-additive codes. Also the singleton bound for SR-additive codes and some results on one weight SR-additive codes are given. Among other important results, we obtain the structure of SR-additive cyclic codes. As some results of the theory, the structure of cyclic ${\mathbb{Z}}_2{\mathbb{Z}}_4$, ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$, ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2)$, $({\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$ and $({\mathbb{Z}}_2+u{\mathbb{Z}}_2)({\mathbb{Z}}_2+u{\mathbb{Z}}_2+v{\mathbb{Z}}_2)$-additive codes are presented.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.801-810
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• 1998

Let A(z), W(z) and C(z) be power series with operator coefficients such that W(z) = A(z)C(z). Let D(A) and D(C) be the state spaces of unitary linear systems whose transfer functions are A(z) and C(z) respectively. Then there exists a Krein space D which is the state space of unitary linear system with transfer function W(z). And the element of D is of the form (f(z) ＋ A(z)h(z), k(z) ＋ C＊(z)g(z)) where (f(z),g(z)) is in D(A) and (h(z),k(z)) is in D(C).

• Kyungpook Mathematical Journal
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• v.64 no.1
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• pp.185-196
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• 2024

In this article, we investigate the growth of meromorphic solutions of $${\alpha}(z)(\frac{{\Delta}_c{\eta}}{{\eta}})^2\,+\,(b_2(z){\eta}^2(z)\;+\;b_1(z){\eta}(z)\;+\;b_0(z))\frac{{\Delta}_c{\eta}}{{\eta}} \atop =d_4(z){\eta}^4(z)\;+\;d_3(z){\eta}^3(z)\;+\;d_2(z){\eta}^2(z)\;+\;d_1(z){\eta}(z)\;+\;d_0(z),$$ where a(z), b_i(z) for i = 0, 1, 2 and d_j (z) for j = 0, ..., 4 are given functions, △_cη = η(z + c) - η(z) with c ∈ ℂ\{0}. In particular, when the a(z), the b_i(z) and the d_j(z) are polynomials, and d₄(z) ≡ 0, we shall show that if η(z) is a transcendental entire solution of finite order, and either deg a(z) ≠ deg d₀(z) + 1, or, deg a(z) = deg d₀(z) + 1 and ρ(η) ≠ ½, then ρ(η) ≥ 1.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.461-468
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• 2004

Let W(z) be a power series with operator coefficients such that multiplication by W(z) is contractive in C(z). The overlapping space $L(\varphi)$ of H(W) in C(z) is a Herglotz space with Herglotz function $\varphi(z)$ which satisfies $\varphi(z)+\varphi^＊(z^{-1})=2[1-W^{*}(z^{-1})W(z)]$. The identity ${}_{L(\varphi)}={-}_{H(W)}$ holds for every f(z) in $L(\varphi)$ and for every vector c.

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.727-749
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• 2015

We obtain the conditions on ${\beta}$ so that $1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3$ implies p(z) ${\prec}$ (2+z)/(2-z), $1+(1-{\alpha})z$, $(1+(1-2{\alpha})z)/(1-z)$, ($0{\leq}{\alpha}$<1), exp(z) or ${\sqrt{1+z}}$. Similar results are obtained by considering the expressions $1+{\beta}zp^{\prime}(z)/p(z)$, $1+{\beta}zp^{\prime}(z)/p^2(z)$ and $p(z)+{\beta}zp^{\prime}(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition ${\mid}log(zf^{\prime}(z)/f(z)){\mid}$ < 1 or ${\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}$ < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1)=0$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.983-1002
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• 2021

We investigate the non-existence of finite order transcendental entire solutions of Fermat-type differential-difference equations [f(z)f'(z)]ⁿ + P²(z)f^m(z + 𝜂) = Q(z) and [f(z)f'(z)]ⁿ + P(z)[∆_𝜂f(z)]^m = Q(z), where P(z) and Q(z) are non-zero polynomials, m and n are positive integers, and 𝜂 ∈ ℂ \ {0}. In addition, we discuss transcendental entire solutions of finite order of the following Fermat-type differential-difference equation P²(z) [f^(k)(z)]² + [αf(z + 𝜂) - 𝛽f(z)]² = e^r(z), where $P(z){\not\equiv}0$ is a polynomial, r(z) is a non-constant polynomial, α ≠ 0 and 𝛽 are constants, k is a positive integer, and 𝜂 ∈ ℂ \ {0}. Our results generalize some previous results.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.363-372
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• 2023

In this paper, we consider two functional equations: (1) h(𝓕(x, y, z) + 2x + y + z) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) + 2x + y) + h(xy) + yh(x + z) + 2h(z), (2) h(𝓕(x, y, z) - y + z + 2e) + 2h(x + y) + h(xy + z) + yh(x) + yh(z) = h(𝓕(x, y, z) - y + 2e) + 2h(x + y + z) + h(xy) + yh(x + z), without any regularity assumption for all x, y, z in a unital algebra A, where 𝓕 : A³ → A is defined by 𝓕(x, y, z) := h(x + y + z) - h(x + y) - h(z) for all x, y, z ∈ A. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.229-246
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• 2024

We consider the delay differential equations $$b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z, w(z))}{Q(z, w(z))}$$, where k ∈ {1, 2}, a(z), b(z) ≢ 0, c(z) ≢ 0 are rational functions, and P(z, w(z)) and Q(z, w(z)) are polynomials in w(z) with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution w with hyper-order ρ2(w) < 1, then either deg_w(P) = deg_w(Q) + 1 ≤ 3 or max{deg_w(P), deg_w(Q)} ≤ 1. In addition, it is shown that in the case max{deg_w(P), deg_w(Q)} = 0 the equations above can have such a solution, with an additional zero density requirement, only if the coefficients of the equation satisfy certain strict conditions.