• Honam Mathematical Journal
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• v.35 no.4
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• pp.717-727
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• 2013

In this paper, we study on a higher order extremal problem relating the Ahlfors map associated to the pair of a finitely connected domain in the complex plane and a point there. We show the power of the Ahlfors map with some error term which is conformally equivalent maximizes any higher order derivative of holomorphic functions at the given point in the domain.

• Honam Mathematical Journal
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• v.27 no.2
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• pp.195-203
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• 2005

We show that the exact Bergman kernel function associated to a $C^{\infty}$ bounded domain in the plane relates the derivatives of the Ahlfors map in an explicit way. And we find several formulas relating the exact Bergman kernel to classical kernel functions in potential theory.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.199-213
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• 2006

We compute the holomorphic derivative of the harmonic measure associated to a $C^\infty$bounded domain in the plane and show that the exact Bergman kernel function associated to a $C^\infty$ bounded domain in the plane relates the derivatives of the Ahlfors map and the Szego kernel in an explicit way. We find several formulas for the exact Bergman kernel and the Szego kernel and the harmonic measure. Finally we survey some other properties of the holomorphic derivative of the harmonic measure.

• Korean Journal of Mathematics
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• v.13 no.2
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• pp.183-191
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• 2005

In this paper we find the Laurent series expansions representing the reproducing kernels. Also we find the number of zeroes of the exact Bergman kernel via parallel slit domain in order to relate the exact Bergman kernel to an extremal problem.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1421-1441
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• 2017

We compute explicitly the matrix represented by the Toeplitz operator on the Hardy space over a smoothly finitely connected bounded domain in the plane with respect to special orthonormal bases consisting of the classical kernel functions for the space of square integrable functions and for the Hardy space. The Fourier coefficients of the symbol of the Toeplitz operator are obtained from zeroth row vectors and zeroth column vectors of the matrix. And we also find some condition for the product of two Toeplitz operators to be a Toeplitz operator in terms of matrices.

• Honam Mathematical Journal
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• v.36 no.3
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• pp.659-668
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• 2014

In this paper, we express the Green function in terms of the classical kernel functions in potential theory. In particular, we obtain a formula relating the Green function and the Szegő kernel function which consists of only the Szegő kernel function in a $C^{\infty}$ smoothly bounded finitely connected domain in the complex plane.

• Honam Mathematical Journal
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• v.36 no.4
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• pp.777-786
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• 2014

We construct an orthonormal basis for the Bergman space associated to a simply connected domain. We use the or-thonormal basis for the Hardy space consisting of the Szegő kernel and the Riemann mapping function and rewrite their area integrals in terms of arc length integrals using the complex Green's identity. And we make a note about the matrix of a Toeplitz operator with respect to the orthonormal basis constructed in the paper.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.447-460
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• 2003

In this paper we study the non-degenerate n-connected canonical domains with n>1 related to the conjecture of S. Bell in [4]. They are connected to the algebraic property of the Bergman kernel and the Szego kernel. We characterize the non-degenerate doubly connected canonical domains.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.335-337
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• 2015

We modify the normalizing constants of the functions in the orthonormal basis for the Bergman space and restate their consequences in the paper that was published in Honam Math. J. 36 (2014), no. 4, pp. 777-786.