A refined discrete Hilbert inequality obtained via the Hermite–Hadamard inequality

In this paper we establish a general method for improving the discrete Hilbert-type inequalities via the Hermite–Hadamard inequality. The general result is then applied to homogeneous kernels. Moreover, some particular homogeneous kernels are considered, and a whole series of refinements of some recent results known from the literature are yielded. In addition some particular non-homogeneous cases are also considered.