• The Pure and Applied Mathematics
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• v.22 no.3
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• pp.263-273
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• 2015

In this paper, a boundary version of Schwarz lemma is investigated. For the function holomorphic f(z) = a + c_pz^p + c_p+₁z^p+1 + ... defined in the unit disc satisfying |f(z) − 1| < 1, where 0 < a < 2, we estimate a module of angular derivative at the boundary point b, f(b) = 2, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.2
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• pp.67-77
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• 2007

This paper is concerned with the proof of so-called Bramble-Hilbert Lemma. We present that $Poincar{\acute{e}}'s$ inequality in [3] implies one of results of Morrey which is crucial in the proof. In this point of view, we recognize that removing the average term in $Poincar{\acute{e}}'s$ inequality fulfills a crucial role in the proof of Bramble-Hilbert Lemma. It is accomplished by adding some polynomial of degree one less than the degree of the Sobolev space in the outset. So, the condition annihilating the set of polynomials $P_{k-1}$ of degree k - 1 is required necessarily in Bramble-Hilbert Lemma.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.543-555
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• 2019

In this paper, we give some results on ${\frac{zf^{\prime}(z)}{f(z)}}$ for the certain classes of holomorphic functions in the unit disc $E=\{z:{\mid}z{\mid}<1\}$ and on ${\partial}E=\{z:{\mid}z{\mid}=1\}$. For the function $f(z)=z^2+c_3z^3+c_4z^4+{\cdots}$ defined in the unit disc E such that $f(z){\in}{\mathcal{A}}_{\alpha}$, we estimate a modulus of the angular derivative of ${\frac{zf^{\prime}(z)}{f(z)}}$ function at the boundary point b with ${\frac{bf^{\prime}(b)}{f(b)}}=1+{\alpha}$. Moreover, Schwarz lemma for class ${\mathcal{A}}_{\alpha}$ is given. The sharpness of these inequalities is also proved.

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.337-345
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• 2023

Different versions of the boundary Schwarz lemma for the 𝒩 (𝜌) class are discussed in this study. Also, for the function g(z) = z+b₂z²+b₃z³+... defined in the unit disc D such that g ∈ 𝒩 (𝜌), we estimate a modulus of the angular derivative of g(z) function at the boundary point 1 ∈ 𝜕D with g'(1) = 1 + 𝜎 (1 - 𝜌), where ${\rho}={\frac{1}{n}}{\sum\limits_{i=1}^{n}}g(c_i)={\frac{g^{\prime}(c_1)+g^{\prime}(c_2)+{\ldots}+g^{\prime}(c_n)}{n}}{\in}g^{\prime}(D)$ and 𝜌≠1, 𝜎 > 1 and c₁, c₂, ..., c_n ∈ 𝜕D. That is, we shall give an estimate below |g"(1)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and z ≠ 0. Estimating is made by using the arithmetic average of n different derivatives g'(c₁), g'(c₂), ..., g'(c_n).

• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.109-114
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• 1995

Before state precisely our main theorem, we want to make some brief historical comment.

• East Asian mathematical journal
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• v.14 no.1
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• pp.99-106
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• 1998

We establish a version of Fernique lemma for Gaussian processes which plays an important role in studying their moduli of continuity properties and related limit theorems.

• The Journal of Korean Association of Computer Education
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• v.12 no.5
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• pp.87-94
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• 2009

This paper presents a Web-based interactive and adaptive learning system that helps students learn the pumping lemma for the family of regular languages. Our system allows the students to proceed with their learning according to their individual differences through Web-Based Instruction and gives them opportunities for the interaction so that they can practice exercise related to the learning and gain feedbacks on the results of the exercises immediately. Especially, the system provides adaptive scaffolding that helps learners understand each step for the proof of the pumping lemma. Unlike existing systems that support learning the pumping lemma, the proposed system defines possible errors in advance and provides appropriate messages for corresponding errors. In addition, the system allows the learners to decompose a string into three parts so that they can understand the pumping lemma precisely.

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.167-170
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• 1993

In this paper, X, X$_{1}$, X$_{2}$, .. will denote any sequence of pairwise independent random variables with common distribution, and b$_{1}$, b$_{2}$.. will denote any sequence of constants. Using Chung [2, Theorem 4.2.5] and the same idea as in Chow and Robbins [1, Lemma 1 and 2] we have the following lemma.

• The Pure and Applied Mathematics
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• v.22 no.2
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• pp.169-178
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• 2015

In this paper, a boundary version of Carathéodory’s inequality is investigated. Also, new inequalities of the Carathéodory’s inequality at boundary are obtained and the sharpness of these inequalities is proved.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.225-232
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• 2003

In this paper, we consider the existence of the solutions to the generalized vector variational-type inequalities for set-valued mappings on Hausdorff topological vector spaces using Fan's geometrical lemma.