Approximation of harmonic functions by algebraic polynomials on a circle of radius smaller than one with constraints on the unit circle

A compact expression is found for the value of the best integral approximation of the linear combination $\lambda P_r+\mu Q_r$, where $P_r$ is the Poisson kernel and $Q_r$ is its conjugate, by trigonometric polynomials of a given order in the form of a combination of the functions $\arctan$ and $\ln$. For $\mu=0$, the expression is Krein's result, and, for $\lambda=0$, it is Nagy's result. If $\lambda\mu\not=0$, the expression is much simpler than the representation in the form of a series found by Bushanskii. It is shown that, if the function of limit values on the unit circle $\Gamma$ of the real part $u=\mathrm{Re}F$ of a certain function $F=u+iv$ that is analytic inside the unit circle and such that $\|u\|_{L(\Gamma)}\le1$ is known, then the problem of the best integral approximation of the linear combination $\lambda u+\mu v$ on a concentric circle of radius $r<1$ by algebraic polynomials is reduced to the integral approximation of the kernel $\lambda P_r+\mu Q_r$ on the period $[0,2\pi)$ by trigonometric polynomials.