  In this paper we establish asymptotically best possible interpolation
Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are
representable as generalized Poisson integrals of the functions $\varphi$ from
the space $L_p$, $1\leq p\leq \infty$. In these inequalities the deviation of
the interpolation Lagrange polynomials $|f(x)- \tilde{S}_{n-1}(f;x)|$ for every
$x\in\mathbb{R}$ is expressed via the best approximations
$E_{n}(\varphi)_{L_{p}}$ of the functions $\varphi$ be trigonometric
polynomials in $L_{p}$-metrics. We also find asymptotic equalities for the
exact upper bounds of points approximations by interpolation trigonometric
polynomials on the classes $C^{\alpha,r}_{\beta,p}$ of generalized Poisson
integrals of the functions, which belong to the unit balls of the spaces $L_p$,
$1\leq p\leq\infty$.
