• Nonlinear Functional Analysis and Applications
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• 제28권4호
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• pp.943-956
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• 2023

Let p(z) be a polynomial of degree n and for any complex number 𝛽, let D_𝛽p(z) = np(z) + (𝛽 - z)p'(z) denote the polar derivative of the polynomial with respect to 𝛽. In this paper, we consider the class of polynomial $$p(z)=(z-z_0)^s \left(a_0+\sum\limits_{{\nu}=0}^{n-s}a_{\nu}z^{\nu}\right)$$ of degree n having a zero of order s at z₀, |z₀| < 1 and the remaining n - s zeros are outside |z| < k, k ≥ 1 and establish upper bound estimates for the maximum of |D_𝛽p(z)| as well as |p(Rz) - p(rz)|, R ≥ r ≥ 1 on the unit disk.

• 대한수학회보
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• 제58권4호
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• pp.915-937
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• 2021

In this paper we deal with the open problem posed by Lü, Li and Yang [10]. In fact, we prove the following result: Let f(z) be a transcendental meromorphic function of finite order having finitely many poles, c₁, c₂, …, c_n ∈ ℂ\{0} and k, n ∈ ℕ. Suppose fⁿ(z), f(z+c₁)f(z+c₂) ⋯ f(z+c_n) share 0 CM and fⁿ(z)-Q₁(z), (f(z+c₁)f(z+c₂) ⋯ f(z+c_n))^(k) - Q₂(z) share (0, 1), where Q₁(z) and Q₂(z) are non-zero polynomials. If n ≥ k+1, then $(f(z+c_1)f(z+c_2)\;{\cdots}\;f(z+c_n))^{(k)}\;{\equiv}\;{\frac{Q_2(z)}{Q_1(z)}}f^n(z)$. Furthermore, if Q₁(z) ≡ Q₂(z), then $f(z)=c\;e^{\frac{\lambda}{n}z}$, where c, λ ∈ ℂ \ {0} such that e^{λ(c₁+c₂+⋯+c_n)} = 1 and λ^k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

• Korean Journal of Mathematics
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• 제7권1호
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• pp.85-101
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• 1999

Let $p$ be analytic in the unit disc U and let $q$ be univalent in U. In addition, let ${\Omega}$ be a set in C and let ${\psi}:c^3{\times}U{\rightarrow}C$. The author determines conditions on ${\psi}$ so that $$\{{\psi}(p(z),zp^{\prime}(z),z^2p^{{\prime}{\prime}}(z);z){\mid}z{\in}U\}{\subset}{\Omega}{\Rightarrow}p(U){\subset}q(U)$$. Applications of this result to differential inequalities, differential subordinations and integral inequalities are presented.

• Progress in Superconductivity
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• 제4권2호
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• pp.109-113
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• 2003

We have developed a planar-type single layer second-order $high-T_{c}$ SQUID gradiometer, which can detect the $d^2$$B_{z/}$dxdy of the second-order field gradient. This SQUID gradiometer consists of four-way 'clover-leaf' pick-up loops and is coupled directly to a 4-junction dc SQUID in such a way that the coupling polarity of the two diagonal loops is opposite to that of the other two loops. The pickup loops are intrinsically balanced for both uniform field and the 1 st-order field gradient. The $YBa_2$$Cu_3$$O_{7}$ thin film was made by pulsed laser deposition method on $SrTiO_3$ single crystal substrate and patterned by photolithography with Ar ion milling technique. Response of this gradiometer was tested for both uniform field and the 2nd-order field gradient. Details of the design, fabrication, and results will be discussed.

• 대한소아치과학회지
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• 제31권4호
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• pp.547-554
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• 2004

본 연구는 복합레진에서 치태 생성 에 영향을 주는 표면 거칠기 및 표면 자유에너지를 나타내는 물방울 접촉각을 측정하고, 치태 부착실험을 통해 두 가지 요소가 미치는 영향에 대하여 평가할 목적으로 시행되었다. 4종의 복합레진(Z-100, Filtek supreme, Durafil, Clearfil AP X)을 비연마군과 연마군으로 각각 5개씩 시편을 제작하였다. 표면 거칠기는 표면조도형상 측정기를, 접촉각은 접촉각측정기를 이용하여 각각 측정하였다. 치태부착 실험은, 시편을 배양액을 넣고 Streptococcus mutans Ingbritt을 접종하여 부착된 치태의 양을 측정하여 평가하였다. 1. 표면조도는 연마되지 않은 대조군에서는 (Z1, DF, CA)>FS의 순 연마된 실험군에서는 CA>Z1>(FS, DF)의 순으로 나타났으며, 실험군이 대조군에 비해 높은 표면 조도를 보였다(p<0.05). 2. 접촉각은 대조군에서는 CA>(FS, DF, Z1)의 순, 실험군에서는 (CA, DF)>(FS, Z1)의 순으로 나타났으며, 실험군이 대조군에 비해 낮은 접촉각을 보였다(p<0.05). 3. 치태침착은 대조군에서는 Z1>(DF, FS)>CA의 순, 실험군에서는 Z1>FS> (CA, DF)의 순으로 나타났으며, 실험군이 대조군에 비해 치태 침착이 더 많았다(p<0.05). 4. 치태 부착도에 대해서 표면 조도는 상관성이 없고, 접촉각은 강한 반비례 관계를 보였다(p<0.05). 즉 치태 부착에 대해 접촉각이 표면 조도 보다 더 큰 관련성을 보였다.

• 대한수학회보
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• 제57권4호
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• pp.839-850
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• 2020

For functions f(z) = z + a₂z² + a₃z³ + ⋯ belonging to particular classes, this paper finds sharp bounds for the initial coefficients a₂, a₃, a₄, as well as the sharp estimate for the second order Hankel determinant H₂(2) = a₂a₄ - a₂³. Two classes are treated: first is the class consisting of f(z) = z + a₂z² + a₃z³ + ⋯ in the unit disk 𝔻 satisfying $$\|\(\frac{z}{f(z)}\)^{1+{\alpha}}\;f^{\prime}(z)-1\|<{\lambda},\;0<{\alpha}<1,\;0<{\lambda}{\leq}1.$$ The second class consists of Bazilevič functions f(z) = z+a₂z²+a₃z³+⋯ in 𝔻 satisfying $$Re\{\(\frac{f(z)}{z}\)^{{\alpha}-1}\;f^{\prime}(z)\}>0,\;{\alpha}>0.$$

• 대한환경공학회지
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• 제39권9호
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• pp.505-512
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• 2017

NaA 제올라이트 분말(Z-PA) 및 펠렛(Z-BA)를 이용하여 수중의 Sr 이온의 흡착특성을 검토하였다. 회분식 실험에서 Z-BA와 Z-PA에 의한 Sr 이온의 흡착은 유사 1차식보다는 유사 2차식에 잘 부합되었고, 등온 흡착실험 결과는 Langmuir 등온식에 잘 부합하였으며, 이 등온식으로부터 구한 최대 흡착량은 Z-PA는 233.32 mg/g, Z-BA는 164.60 mg/g이었다. 연속식 실험에서 Sr 이온의 농도가 증가함에 따라 Sr 이온의 총 흡착량(q)는 증가하였으나 파과시간, 처리 부피($V_{eff}$) 및 총 제거율(R)은 감소하였다. 실험으로부터 구한 파과곡선은 Thomas 모델식에 의해 잘 모사되었다.

• 대한수학회지
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• 제45권1호
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• pp.63-78
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• 2008

We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.

• 호남수학학술지
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• 제38권4호
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• pp.701-724
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• 2016

For $f_i$ belonging to various subclasses of univalent functions, we investigate the product given by $h(z)=z{\prod_{i=1}^{n}}(f_i(z)/z)^{{\gamma}_i}$.The largest radius ${\rho}$ is determined such that $h({\rho}z)/{\rho}$ is starlike of order ${\beta}$, $0{\leq}{\beta}$ < 1 or to belong to other subclasses of univalent functions. We also determine the sharp radius of starlikeness of order ${\beta}$and other radius for the convolution f*g of two starlike functions f, g.

• 대한수학회보
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• 제47권3호
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• pp.443-454
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• 2010

In this paper, we investigate the zeros distribution and Borel direction for the solutions of linear homogeneous differential equation $f^{(n)}+A_{n-2}(z)f^{(n-2)}+{\cdots}+A_1(z)f'+A_0(z)f=0(n{\geq}2)$ in an angular domain. Especially, we establish a relation between a cluster ray of zeros and Borel direction.