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The authors consider the optimization problem related to the integral representation of the deviation of positive linear operators on the classes of (ψ, β )-differentiable functions in the integral metric. The Poisson integral is taken as a positive linear operator, being the solution of the Laplace equation in polar coordinates with the corresponding initial conditions given on the boundary of the unit disc. The Poisson integral refers to operators with a delta-like kernel; therefore, it is the best apparatus for solving many problems of applied mathematics, namely: optimization methods and calculus of variations, mathematical control theory, theory of dynamical systems and game problems of dynamics, applied nonlinear analysis and moving objects search. The classes of (ψ, β )-differentiable functions are generalizations of the well-known Sobolev, Weyl–Nagy, etc. classes in optimization problems, on which the asymptotic properties of solutions of Laplace equations in the unit disc are analyzed. The problem solved in the paper will make it possible to generate high-quality mathematical models of many natural and social processes. [ABSTRACT FROM AUTHOR]