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The positivity of the integrand of certain Euclidean space functionalintegrals implies that the free energy per unit volume for QCD with a baryonchemical potential $\mu_B$ (and zero isospin chemical potential) is greaterthan the free energy with isospin chemical potential $\mu_I = (2 \mu_B)/N_c$(and zero baryon chemical potential). This relation implies a bound on thenucleon mass: there exists a particle X in QCD such that $M_N \ge (N_c m_X)/(2I_X)$ where $m_X$ is the mass and of the particle and $I_X$ is its isospin.Comment: Four pages. Version to appear in prin

QCD Inequalities for the Nucleon Mass and the Free Energy of Baryonic Matter