We propose new bounds on black hole entropy in terms of the specific heat at fixed charges. For special spherically symmetric and static black holes ($g_{tt}{g}_{rr}=-1$), we prove the bounds with suitable sufficient energy conditions. For more general static and rotating black holes, we test the bounds with a variety of examples. This work extends the previously known algebraic inequalities of the black hole thermodynamic variables such as the Penrose inequality, to include their derivatives.

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• Classical black holes
• General relativity