• Communications for Statistical Applications and Methods
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• v.11 no.3
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• pp.455-463
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• 2004

In this paper, we derive sharp upper and lower expectation bounds on the extreme order statistics from possibly dependent random variables whose marginal distributions are only known. The marginal distributions of the considered random variables may not be the same and the expectation bounds are completely determined by the marginal distributions only.

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.389-404
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• 2023

We prove sharp bounds for Hankel determinants for starlike functions f with respect to symmetrical points, i.e., f given by $f(z)=z+{\sum{_{n=2}^{\infty}}}\,{\alpha}_nz^n$ for z ∈ 𝔻 satisfying $$Re{\frac{zf^{\prime}(z)}{f(z)-f(-z)}}>0,\;z{\in}{\mathbb{D}}$$. We also give sharp upper and lower bounds when the coefficients of f are real.

• Structural Engineering and Mechanics
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• v.13 no.3
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• pp.299-312
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• 2002

In this paper, a new method to solve the dynamic response problem for structures with interval parameters is presented. It is difficult to obtain all possible solutions with sharp bounds even an optimum scheme is adopted when there are many interval structural parameters. With the interval algorithm, the expressions of the interval stiffness matrix, damping matrix and mass matrices are developed. Based on the matrix perturbation theory and interval extension of function, the upper and lower bounds of dynamic response are obtained, while the sharp bounds are guaranteed by the interval operations. A numerical example, dynamic response analysis of a box cantilever beam, is given to illustrate the validity of the present method.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.785-800
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• 2021

In this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for |f'(0)| and sharp lower bounds for |f'(c)| with c ∈ ∂D = {z : |z| = 1}. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function f(z) from below according to the second Taylor coefficients of f about z = 0 and z = z₀ ≠ 0. Thanks to these inequalities, we see the relation between |f'(0)| and 𝕽f(0). Similarly, we see the relation between 𝕽f(0) and |f'(c)| for some c ∈ ∂D. The sharpness of these inequalities is also proved.

• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.461-468
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• 2023

This paper attempts to investigate a new subfamily 𝓢𝓣_𝜗,𝜎 (𝛼, 𝛽, 𝛾, 𝜇) of spirallike functions endowed with Mittag-Leffler and Wright functions. The paper further investigates sharp coefficient bounds for functions that belong to this class.

• Korean Journal of Mathematics
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• v.26 no.1
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• pp.103-127
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• 2018

This paper is dedicated to studying some Hausdorff operators on the Heisenberg group ${\mathbb{H}}^n$. The sharp bounds on the strong-type weighted Lorentz spaces ${\Lambda}^p_u(w)$ and the weak-type weighted Lorentz spaces ${\Lambda}^{p,{\infty}}_u(w)$ are investigated. Especially, the results cover the classical power weighted space $L^{p,q}_{\alpha}$. The results are also extended to the product spaces ${\Lambda}^{p_1}_{u_1}(w_1){\times}{\Lambda}^{p_2}_{u_2}(w_2)$, especially for $L^{p_1,q_1}_{{\alpha}_1}{\times}L^{p_2,q_2}_{{\alpha}_2}$. Our proofs are quite different from those in previous documents since the duality principle, and some well-known inequalities concerning the weights are adopted. The results recover the existing results as well as we obtain new results in the new and old settings.

• East Asian mathematical journal
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• v.30 no.3
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• pp.355-360
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• 2014

Let D be a finite digraph with the vertex set V (D) and the arc set A(D). A function f : $V(D){\rightarrow}\{-1,\;1\}$ defined on the vertices of a digraph D is called a bad function if $f(N^-(v)){\leq}1$ for every v in D. The weight of a bad function is $f(V(D))=\sum\limits_{v{\in}V(D)}f(v)$. The maximum weight of a bad function of D is the the negative decision number ${\beta}_D(D)$ of D. Wang [4] studied several sharp upper bounds of this number for an undirected graph. In this paper, we study sharp upper bounds of the negative decision number ${\beta}_D(D)$ of for a digraph D.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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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.$$

• Structural Engineering and Mechanics
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• v.44 no.1
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• pp.73-84
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• 2012

This paper strongly addresses to the problem of the mechanical systems in which parameters are uncertain and bounded. Interval calculation is used to find sharp bounds of the structural parameters for infilled frame system modeled with finite element method. Infill walls are generally treated as non-structural elements considerably to improve the lateral stiffness, strength and ductility of the structure together with the frame elements. Because of their complex nature, they are often neglected in the analytical model of building structures. However, in seismic design, ignoring the effect of infill wall in a numerical model does not accurately simulate the physical behavior. In this context, there are still some uncertainties in mechanical and also geometrical properties in the analysis and design procedure of infill walls. Structural uncertainties can be studied with a finite element formulation to determine sharp bounds of the structural parameters such as wall thickness and Young's modulus. In order to accomplish this sharp solution as much as possible, interval finite element approach can be considered, too. The structural parameters can be considered as interval variables by using the interval number, thus the structural stiffness matrix may be divided into the product of two parts which correspond to the interval values and the deterministic value.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1791-1809
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• 2018

Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and $T_q(q{\in}(1,{\infty}))$ be the vector-valued operator defined by $T_qf(x)=({\sum}_{k=1}^{\infty}{\mid}T\;f_k(x){\mid}^q)^{1/q}$. In this paper, by proving certain weak type endpoint estimate of L log L type for the grand maximal operator of T, the author establishes some quantitative weighted bounds for $T_q$ and the corresponding vector-valued maximal singular integral operator.