  Let (X,L) be a polarized K\"ahler manifold that admits an extremal K\"ahler
metric in c1(L). We show that on a nearby polarized deformation that preserves
the symmetry induced by the extremal vector field of (X,L), the modified
K-energy is bounded from below. This generalizes a result of Chen,
Sz\'ekelyhidi and Tosatti to extremal metrics. Our proof also extends a
convexity inequality on the space of K\"ahler potentials due to X.X. Chen to
the extremal metric setup. As an application, we compute explicit polarized
4-points blow-ups of CP1\times CP1 that carry no extremal metric but with
modified K-energy bounded from below.
