On an integral inequality

In this paper we deduce an integral inequality which is an analog of a known two-parameter inequality of Hardy and Littlewood ([1], Theorem 382). A need for inequalities of a similar type arises, for example, in studying the imbedding of the functional spaces $B_{p,\,\theta}^l$ in the space $L_q$ if this study leads to a basis of the method of integral representations of functions.