Integral estimates for derivatives of analytic functions outside convex domains

In the present paper weight integral estimates are obtained for derivatives of functions which are analytic in the exterior of convex bounded domains. The estimates are obtained in terms of integrals of functions vanishing at infinity. This result generalizes the Hardy–Littlewood theorem for exteriors of convex bounded domains. Theorems of this kind have been earlier obtained by K. P. Isaev and R. S. Yulmukhametov for the power weight and for the first derivative of an analytic function of the first order belonging to $L^2$. N. M. Tkachenko and F. A. Shamoyan have generalized this result for all higher order derivatives belonging to the space $L^p$. In the present paper the class of weights under consideration is essentially enlarged.