  Estimating the cosmic-ray acceleration efficiency $ \epsilon $ in supernova
remnants (SNRs) through observations is a challenging task in general. Based on
the Rankine-Hugoniot shock conditions, we find an anticorrelation between $
\epsilon $ and the power-law spectral index $ \alpha $ of relativistic particle
distribution produced via diffusive particle acceleration by nonrelativistic
shocks, implying more efficient acceleration in older SNRs with harder radio
spectra. Then $ \epsilon $ may be estimated from some hard radio spectral index
measurements. Assuming the particle distribution in downstream of strong shocks
to be a nonrelativistic Maxwellian plus a relativistic power law with a
high-energy cutoff, we also find that the injection rate for relativistic
particles $ \eta $ needs to $ \gtrsim 10^{-6} $ for a prominent decrease of the
adiabatic index in SNRs, which implies higher compression ratio and lower
values of $ \alpha $. This threshold of $ \eta $ increases with the shock speed
$ u_1 $, which may explain the relatively harder radio spectra of older SNRs
with lower $ u_1 $. We show that $ \eta $ and/or the relativistic cutoff
momentum $ p_\text{m} $ need to be low for old SNRs, and expect a gradual
increase of $ \epsilon $ as SNR evolves with gradually decreasing $ \eta $ and
$ p_\text{m} $.
