  In this paper, we give a lower bound estimate for the diameter of a
Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an
analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a
Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li
about the non-existence of compact self-expanders in a Euclidean space.
