Temperature and energy derivatives of Doppler broadening functions by Steffensen's inequality technique

Bounds of asymmetric Doppler broadening function χ(x, θ) are obtained ab initio from application of Steffensen's inequality. It is shown how these bounds and their averages provide approximations for the function to any desired accuracy. These bounds along with similar bound approximations for ψ(x, θ) lead to bounds for temperature and energy derivatives of Doppler broadening functions. Accuracies of the approximations for the derivatives are also discussed. It is demonstrated that calculation of derivatives to desired accuracies require ψ and χ functions evaluated with higher relative accuracies.