By integrating the linearized Vlasov equation, the author derives an integral form of the constitutive relation (the relation between the h.f. electric field E and the h.f. current J) valid to all orders in the Larmor radius in inhomogeneous plasmas with gradients perpendicular to the static magnetic field. It has the the form of a convolution integral in Fourier kperpendicular to -space, whose kernel can be expanded in Bessel functions, or put in a form analogous to the Gordeev integral form of the usual uniform plasma conductivity tensor, to which it reduces in the uniform plasma limit. Alternatively, it can be formulated as an integral equation in real space. Both formulations can be useful for the investigation of ion Bernstein waves near ion cyclotron resonances and near the lower hybrid resonance, where a WKB analysis alone is insufficient.