The global maximum of an entropy function with different decision levels for a threelevel scalar quantizer performed after a discrete wavelet transform was derived. Herein, we considered the case of entropyconstrained scalar quantization capable of avoiding many compression ratio reductions as the mean squared error was minimized. We also dealt with the problem of minimum entropy with an error bound, which was referred to as the rate distortion function. For generalized Gaussian distributed input signals, the Shannon bound would decrease monotonically when the parameter of distribution 7 was to leave from 2. That is, the Gaussian distributions would contain the highest Shannon bound among the generalized Gaussian distributions. Additionally, we proposed two numerical approaches of the secant and false position methods implemented in real cases to solve the problems of entropyconstrained scalar quantization and minimum entropy with an error bound. The convergence condition of the secant method was also addressed.