  The Gibbs-Bogoliubov inequality states that the free energy of a system is
always lower than that calculated by a trial function. In this study, we show
that a counterpart of the Gibbs-Bogoliubov inequality holds on the Nishimori
line for Ising spin-glass models with Gaussian randomness. Our inequality
states that the quenched free energy of a system is always lower than that
calculated using a quenched trial function. The key component of the proof is
the convexity of the pressure function $\mathbb{E}\left[\log Z_{} \right]$ with
respect to the parameters along the Nishimori line, which differs from the
conventional convexity with respect to the inverse temperature. When our
inequality was applied to mean-field models, such as the
Sherrington-Kirkpatrick model and $p$-spin model, the bound coincided with the
replica-symmetric solution indicating that the equality holds.
