• Honam Mathematical Journal
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• v.44 no.1
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• pp.135-145
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• 2022

This paper aims to investigate characterizations on parameters k₁, k₂, k₃, k₄, k₅, l₁, l₂, l₃, and l₄ to find relation between the class of 𝓗(k, l, m, n, o) hypergeometric functions defined by $$5_F_4\[{\array{k_1,\;k_2,\;k_3,\;k_4,\;k_5\\l_1,\;l_2,\;l_3,\;l_4}}\;:\;z\]=\sum\limits_{n=2}^{\infty}\frac{(k_1)_n(k_2)_n(k_3)_n(k_4)_n(k_5)_n}{(l_1)_n(l_2)_n(l_3)_n(l_4)_n(1)_n}z^n$$. We need to find k, l, m and n that lead to the necessary and sufficient condition for the function zF([W]), G = z(2 - F([W])) and $H_1[W]=z^2{\frac{d}{dz}}(ln(z)-h(z))$ to be in 𝓢^*(2^-r), r is a positive integer in the open unit disc 𝒟 = {z : |z| < 1, z ∈ ℂ} with $$h(z)=\sum\limits_{n=0}^{\infty}\frac{(k)_n(l)_n(m)_n(n)_n(1+\frac{k}{2})_n}{(\frac{k}{2})_n(1+k-l)_n(1+k-m)_n(1+k-n)_nn(1)_n}z^n$$ and $$[W]=\[{\array{k,\;1+{\frac{k}{2}},\;l,\;m,\;n\\{\frac{k}{2}},\;1+k-l,\;1+k-m,\;1+k-n}}\;:\;z\]$$.

• 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.

• Kyungpook Mathematical Journal
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• v.28 no.2
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• pp.129-139
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• 1988

Let $T^{\alpha}_{\lambda}$(p, A, B) denote the class of functions $$f(z)=z^p-{\sum\limits^{\infty}_{k=1}}{\mid}a_{p+k}{\mid}z^{p+k}$$ which are regular and p valent in the unit disc U = {z: |z| <1} and satisfying the condition $\left|{\frac{{e^{ia}}\{{\frac{f^{\prime}(z)}{z^{p-1}}-p}\}}{(A-B){\lambda}p{\cos}{\alpha}-Be^{i{\alpha}}\{\frac{f^{\prime}(z)}{z^{p-1}}-p\}}}\right|$<1, $z{\in}U$, where 0<${\lambda}{\leq}1$, $-\frac{\pi}{2}$<${\alpha}$<$\frac{\pi}{2}$, $-1{\leq}A$<$B{\leq}1$, 0<$B{\leq}1$ and $p{\in}N=\{1,2,3,{\cdots}\}$. In this paper, we obtain sharp results concerning coefficient estimates, distortion theorem and radius of convexity for the class $T^{\alpha}_{\lambda}$(p, A, B). It is further shown that the class $T^{\alpha}_{\lambda}$(p, A, B) is closed under "arithmetic mean" and "convex linear combinations". We also obtain class preserving integral operators of the form $F(z)=\frac{p+c}{z^c}{\int^z_0t^{c-1}}f(t)dt$, c>-p, for the class $T^{\alpha}_{\lambda}$(p, A, B). Conversely when $F(z){\in}T^{\alpha}_{\lambda}$(p, A, B), radius of p valence of f(z) has also determined.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.141-148
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• 1995

In this paper we shall consider function p(z) analytic in the open unit circle D and the solutions y(z) of the differential equation y"(Z) ＋ p(z)y(z) = 0. (1.1) The ratio f(z) = u(z)/v(z) of any two independent solutions u(z) and v(z) of (1.1) will be function f(z), meromorphic in D with only simple poles, and such that f'(z) (equation omitted) 0. We shall say that a meromorphic function which satisfies these two condition belongs to the restricted class.(omitted)

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.331-357
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• 2020

Let p be an analytic function defined on the open unit disk 𝔻. We obtain certain differential subordination implications such as ψ(p) := p^λ(z)(α+βp(z)+γ/p(z)+δzp'(z)/p^j(z)) ≺ h(z) (j = 1, 2) implies p ≺ q, where h is given by ψ(q) and q belongs to 𝒫, by finding the conditions on α, β, γ, δ and λ. Further as an application of our derived results, we obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy |log(zf'(z)/f(z))| < 1, |(zf'(z)/f(z))² - 1| < 1 and zf'(z)/f(z) lying in the parabolic region v² < 2u - 1.

• Honam Mathematical Journal
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• v.21 no.1
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• pp.57-64
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• 1999

Let K be a algebraically closed field of characteristic 0 and G be abelian group $Z_2{\times}Z_2{\times}Z_2$ or $Z_2{\times}Z_4$. We find the conditions which the elements of the group ring KG are unit and idempotent respecting using the basic table matrix of G. We can see that if ${\alpha}={\sum}r(g)g$ is an idempotent element of KG, then $r(1)=0,\;\frac{1}{{\mid}G{\mid}},\;\frac{2}{{\mid}G{\mid}},\;{\cdots},\frac{{\mid}G{\mid}-1}{{\mid}G{\mid}},\;1$.

• Korean Journal of Medicinal Crop Science
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• v.19 no.1
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• pp.1-8
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• 2011

Biological characteristics of 5 Zanthoxylum schinifolium (Zs) fruits such as Z1 (early August), Z2 (middle August), Z3 (middle September), Z4 (early October) and Z5 (middle October) according to harvesting time were evaluated. As fruits ripened, average weight of Zs increased from 4.8mg (Z1) to 50.7mg (Z5), while moisture contents decreased from 74.6% (Z1) to 55.2% (Z5). Crude fat contents of the fruits during ripening increased from 1% (Z1) to 10.6% (Z5). The major fatty acids in Zs were palmitic (C16:0), palmitoleic (C16:1), oleic (C18:1), and linoleic (C18:2) acids. Linoleic acid (C18:2) was a main fatty acid in Z1 and Z2, whereas oleic acid (C18:1) was found as a main one in the other Zs. The ratio of unsaturated fatty acid to total fatty acids increased from 60% (Z1) to 80% (Z3~Z5) during ripening. Among ripening stages, Z4 had the highest contents of total fatty acids ($3,355{\mu}g/g$) and total unsaturated fatty acids ($2,753{\mu}g/g$). Forty six volatile compounds in Zs were also identified. The major volatile compounds were ${\alpha}-pinene$, ${\beta}-myrcene$, ${\beta}-ocimene$, 2-nonanone, estragole, 2-undecanone, and ${\beta}-caryophyllene$. Major volatile components of Z1 were ${\beta}-ocimene$ (20.8 peak area %) and ${\alpha}-pinene$ (9.7 peak area %). In Z2, estragole (30.1 peak area %) was a main volatile compound, but the contents of ${\alpha}-pinene$ (0.4 peak area %), ${\beta}-myrcene$ (0.3 peak area %), and ${\beta}-ocimene$ (0.6 peak area %) were lower than those in Z1. Especially, estragole used as perfumes and as a food additive for flavor was drastically increased to 91.2 (Z3) and 92% (Z4) as fruits ripened.

• Journal of Korean Society of Environmental Engineers
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• v.31 no.11
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• pp.941-946
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• 2009

Two types of synthetic zeolites, commercially used (Z-WK) and synthesized by coal fly ash (Z-C1), and raw coal fly ash(F-C1) were examined for its kinetics and adsorption capacities of cobalt. Experimental data are fitted with kinetic models, Lagergen $1^{st}$ and $2^{nd}$ order models, and four types of adsorption isotherm models, Langmuir, Freundlich, Redlich-Peterson, and Koble-Corrigan. Synthesized zeolite (Z-C1) which had 1.51 of Si/Al ratio was synthesized by raw coal fly ash from a thermal power plant. Adsorption capacities with three types of adsorbents, Z-WK, Z-C1, and F-C1, were in the order of Z-C1 (94.15 mg/g) > F-C1 (92.94 mg/g) > Z-WK (88.56mg/g). The adsorption kinetics of Z-WK and Z-C1 with cobalt could be accurately described by a pseudo-second-order rate equation. The adsorption isotherms of Z-WK and Z-C1 with cobalt were well fitted by the Langmuir and Redlich-Peterson equation. Z-C1 will be used to remove cobalt in water as a more efficient absorbent.

• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.959-968
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• 2024

Suppose f belongs to the Bloch space with f(0) = 0. For 0 < r < 1 and 0 < p < ∞, we show that $$M_p(r,\,f)\,=\,({\frac{1}{2\pi}}{\int_{0}^{2\pi}}\,{\mid}f(re^{it}){\mid}^pdt)^{1/p}\,{\leq}\,({\frac{{\Gamma}(\frac{p}{2}+1)}{{\Gamma}(\frac{p}{2}+1-k)}})^{1/p}\,{\rho}{\mathcal{B}}(log\frac{1}{1-r^2})^{1/2},$$ where ρʙ(f) = sup_z∈ⅅ(1 - |z|²)|f'(z)| and k is the integer satisfying 0 < p - 2k ≤ 2. Moreover, we prove that for 0 < r < 1 and p > 1, $${\parallel}f_r{\parallel}_{B_q}\,{\leq}\,r\,{\rho}{\mathcal{B}}(f)(\frac{1}{(1-r^2)(q-1)})^{1/q},$$ where f_r(z) = f(rz) and ||·||ʙ_q is the Besov seminorm given by ║f║ʙ_q = (∫_𝔻 |f'(z)|^q(1-|z|²)^q-2dA(z)). These results improve previous results of Clunie and MacGregor.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.991-1002
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• 2020

We show that when n > m, the following delay differential equation fⁿ(z)f'(z) + p(z)(f(z + c) - f(z))^m = r(z)e^q(z) of rational coefficients p, r doesn't admit any transcendental entire solutions f(z) of finite order. Furthermore, we study the conditions of α₁, α₂ that ensure existence of transcendental meromorphic solutions of the equation fⁿ(z) + f^n-2(z)f'(z) + P_d(z, f) = p₁(z)e^α₁(z) + p₂(z)e^α₂(z). These results have improved some known theorems obtained most recently by other authors.