We examine the entropy generation regarding its magnitude and the limit as time tends to infinity and apply the second law of thermodynamics to develop mathematical inequalities with heat conduction in adiabatic cylinders. The former shows a bounded entropy generation if the heat conduction is initiated by the initial temperature distribution, but unbounded if the heat conduction involves a heat source with positive volume average over the cylinder. The latter yields various innovative relations that are useful both for studying differential equations and for examining accuracy of analytical, numerical and experimental results. The work not only builds up the relation between the second law of thermodynamics and mathematical inequalities, but also offers some fundamental insights of universe and our future. (C) 2018 Elsevier Ltd. All rights reserved.