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Riemann sums, a classical method for approximating the definite integral of afunction, have been extensively studied in the past. However, their monotonicproperties, while still of great importance, particularly in approximationtheory and interpolation theory, remain somewhat obscure. This paper isdedicated to proving general theorems about the monotonicity of left and rightRiemann sums, a problem first raised by Fej\'er in 1950. We provide amuch-needed review of the literature on the problem and offer several newsufficient and necessary conditions for the monotonicity of Riemann sums.Additionally, we present a new insightful proof of a fundamental theoremrelated to these sums using tools from the theory of majorization. The authoralso delves deeper into a question posed by Borwein, almost resolving itcompletely.

On the monotonicity of left and right Riemann sums