  Due to the intrinsically initial singularity of solution and the discrete
convolution form in numerical Caputo derivatives, the traditional $H^1$-norm
analysis (corresponding to the case for a classical diffusion equation) to the
time approximations of a fractional subdiffusion problem always leads to
suboptimal error estimates (a loss of time accuracy). To recover the
theoretical accuracy in time, we propose an improved discrete Gr\"{o}nwall
inequality and apply it to the well-known L1 formula and a fractional
Crank-Nicolson scheme. With the help of a time-space error-splitting technique
and the global consistency analysis, sharp $H^1$-norm error estimates of the
two nonuniform approaches are established for a reaction-subdiffusion problems.
Numerical experiments are included to confirm the sharpness of our analysis.
