A Holographic Proof of R\'enyi Entropic Inequalities

We prove R'enyi entropic inequalities in a holographic setup based on the recent proposal for the holographic formula of R'enyi entropies when the bulk is stable against any perturbation. Regarding the R'enyi parameter as an inverse temperature, we reformulate the entropies in analogy with statistical mechanics, which provides us a concise interpretation of the inequalities as the positivities of entropy, energy and heat capacity. This analogy also makes clear a thermodynamic structure in deriving the holographic formula. As a by-product of the proof we obtain a holographic formula to calculate the quantum fluctuation of the modular Hamiltonian. A few examples of the capacity of entanglement are examined in detail.