Zipf–Mandelbrot and Shannon entropies are some basic and useful tools to quantify information about certain phenomena in various fields of science and technology, for example statistics, ecology, biology, and information theory. In this paper, we obtain some new bounds for generalized Shannon and Zipf–Mandelbrot entropies by using some specific refinements of Jensen’s inequality. Then, as a consequence of these bounds, we deduce some new bounds for Zipf–Mandelbrot and Shannon entropies. Finally, we demonstrate the sharpness of the proposed bounds through numerical experiments.