Upper bound for the average entropy production based on stochastic entropy extrema

The second law of thermodynamics, which asserts the non-negativity of the average total entropy production of a combined system and its environment, is a direct consequence of applying Jensen's inequality to a fluctuation relation. It is also possible, through this inequality, to determine an upper bound of the average total entropy production based on the entropies along the most extreme stochastic trajectories. In this work, we construct an upper bound inequality of the average of a convex function over a domain whose average is known. When applied to the various fluctuation relations, the upper bounds of the average total entropy production are established. Finally, by employing the result of Neri, Rold\'an, and J\"ulicher [Phys. Rev. X 7, 011019 (2017)], we are able to show that the average total entropy production is bounded only by the total entropy production supremum, and vice versa, for a general non-equilibrium stationary system.