Steffensen's inequality is used to obtain new properties of nuclear Doppler broadening functions. We apply the inequality on subinterval integrals of these functions to obtain bounds that provide new approximations for the Doppler broadening functions. The Taylor series is used to further simplify the analytic approximations for the bounds to sums of terms of elementary transcendental functions. The approximations for bounds are able to reproduce the functions with any desired decimal place accuracy. The average of the lower and upper bounds provide better approximations to achieve the same level of decimal place accuracy and are much more efficient computationally. The method is capable of computing the functions to arbitrary accuracy as the inequality essentially gives the bounds of the functions.