For a Banach space B of functions which satisfies for some m>0
a significant improvement for lower estimates of the moduli of smoothness ωr(f,t)B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ℝd or for which translations are isometries or on Sd−1 for which rotations are isometries. Results for C0 semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An Lp space with 1<p<∞ satisfies () where s=max (p,2), and many Orlicz spaces are shown to satisfy () with appropriate s.