  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.
