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Jensen’s inequality is one of the fundamental inequalities which has several applications in almost every field of science. In 2003, Mercer gave a variant of Jensen’s inequality which is known as Jensen–Mercer’s inequality. The purpose of this article is to propose new bounds for Csiszár and related divergences by means of Jensen–Mercer’s inequality. Also, we investigate several new bounds for Zipf–Mandelbrot entropy. The idea of this article may further stimulate research on information theory with the help of Jensen–Mercer’s inequality.