Let G be a connected graph of order n. The remoteness of G, denoted by ρ, is the maximum average distance from a vertex to all other vertices. Let partial_{1}geqcdotsgeqpartial_{n}, partial_{1}^{L}geqcdotsgeqpartial_{n}^{L} and partial_{1} ^{Q}geqcdotsgeqpartial_{n}^{Q} be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on rho+partial _{1}, rho-partial_{n}, rho+partial_{1}^{L}, partial_{1} ^{L}-rho, 2rho+partial_{1}^{Q} and partial_{1}^{Q}-2rho and the corresponding extremal graphs are also characterized.