In this paper, by using gradients of Gateaux differentiable G-increasing functions, we establish some interpolation results for the following G-majorization triangle inequality: (x+y)↓≺Gx↓+y↓ for vectors x,y of an Eaton triple with normal map (⋅)↓. In particular, we show refinements for the majorization Ky Fan's eigenvalue inequality λ(x+y)≺mλ(x)+λ(y) for x,y∈Hn, where Hn is the space of n×n Hermitian matrices and λ(⋅) is the eigenvalue operator on Hn. We derive analogous results for an inner product space with its standard norm, and for real skew-symmetric matrices with their singular values.