On polinomial approximation of entire transcendental functions

M.N. Sheremeta introduced the notion of the $\alpha$-order $\rho_{\alpha}$ so that to extend the scale of growth of maximum modulus of entire transcendental functions, having the order zero $\rho=0$. He established the relation of Hadamard type too. In this article the limiting equalities, connecting the indicated characteristic of an entire function with the sequence of its best polynomial approximation in some Banach spaces of functions analytic in the unit disk have been obtained.