Geodesic
Approximation of Subharmonic Functions in the Half-Plane by the Logarithm of the Modulus of an Analytic Function
Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 483-492.

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We approximate subharmonic functions defined in the open half-plane in the uniform metric outside the exceptional set by the logarithm of the modulus of an analytic (in the half-plane) function for the cases of a finite order and of an infinite lower order. We also obtain an estimate for the size of the exceptional set. It is shown that, in the case of a finite order, the obtained accuracy of the approximation cannot be essentially improved. 
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M. A. Hirnyk. Approximation of Subharmonic Functions in the Half-Plane by the Logarithm of the Modulus of an Analytic Function. Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 483-492. http://geodesic.mathdoc.fr/item/MZM_2005_78_4_a0/

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