• Korean Journal of Mathematics
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• v.30 no.3
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• pp.525-532
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• 2022

Let 𝓟_n denote the space of all complex polynomials $P(z)=\sum\limits_{j=0}^{n}{\alpha}_jz^j$ of degree n. Let P ∈ 𝓟_n, for any complex number α, D_αP(z) = nP(z) + (α - z)P'(z), denote the polar derivative of the polynomial P(z) with respect to α and B_n denote a family of operators that maps 𝓟_n into itself. In this paper, we combine the operators B and D_α and establish certain operator preserving inequalities concerning polynomials, from which a variety of interesting results can be obtained as special cases.

• Bulletin of the Korean Mathematical Society
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• v.45 no.1
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• pp.1-10
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• 2008

We prove a sharp inequality for some multilinear operator related to Marcinkiewicz integral operator. As application, we obtain the weighted norm inequality and L log L type estimate for the multilinear operator.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.15-26
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• 2002

In this paper, we suggest and analyze a class of resolvent dynamical systems associated with mixed variational inequalities. We study the globally asymptotic stability of the solution of the resolvent dynamical systems for the pseudomonotone operators. We also discuss some special cases, which can be obtained from our main results.

• Korean Journal of Mathematics
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• v.27 no.1
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• pp.165-173
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• 2019

In this paper, the definition b-numerical radius and b-norm is introduced and we present several b-numerical radius inequalities. Some applications of these inequalities are considered as well.

• East Asian mathematical journal
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• v.18 no.1
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• pp.95-109
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• 2002

In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.191-200
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• 2024

Motivated by the previously reported results, this work attempts to provide fresh refinements to both vector and numerical radius inequalities by providing a refinement to the well known Buzano's inequality which as a consequence yielded another refinement of the Cauchy-Schwartz (CS) inequality. Utilizing the new refinements of the Buzano's and Cauchy-Schwartz inequalities, we proceed to obtain various vector and numerical radius type inequalities. Methods used in the paper are standard for the operator theory inequality topics.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.169-182
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• 2011

The main object of this paper is to introduce and investigate new subclasses of normalized analytic functions in the open unit disc $\mathbb{U}$, which generalize the familiar class of k-starlike functions. The various properties and characteristics for functions belonging to these classes derived here include (for example) coefficient inequalities, distortion theorems involving fractional calculus, extreme points, integral operators and integral means inequalities.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.757-780
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• 2022

In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights ω, ρ, φ and ψ to hold the following weak type modular inequality $${\mathcal{U}}^{-1}\({\int_{{\mid}{\mathcal{I}}f{\mid}>{\gamma}}}\;{\mathcal{U}}({\gamma}{\omega}){\rho}\){\leq}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}(C{\mid}f{\mid}{\phi}){\psi}\),$$ where ${\mathcal{I}}$ is the modified integral Hardy operators. We also obtain a necesary and sufficient condition for the following extra-weak type integral inequality $${\omega}\(\{{\left|{\mathcal{I}}f\right|}>{\gamma}\}\){\leq}{\mathcal{U}}{\circ}{\mathcal{V}}^{-1}\({\int}_{0}^{\infty}{\mathcal{V}}\(\frac{C{\mid}f{\mid}{\phi}}{{\gamma}}\){\psi}\).$$ Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operator and its integral version.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.109-120
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• 2023

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗 (Ω) over some set Ω with the reproducing kernel k_λ is defined by $${\tilde{A}}(\lambda)=\,\;{\lambda}{\in}{\Omega}$$. The Berezin number of an operator A is defined by $$ber(A):=\sup_{{\lambda}{\in}{\Omega}}{\mid}{\tilde{A}}({\lambda}){\mid}$$. We study some problems of operator theory by using this bounded function Ã, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) ≤ ber (A) ber (B), for some operators A and B on 𝓗 (Ω). Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.

• Kyungpook Mathematical Journal
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• v.31 no.2
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• pp.193-200
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• 1991