  It was proved in Chen's paper \cite{chen} that every real hypersurface in the
complex projective plane of constant holomorphic sectional curvature $4$
satisfies $$ \delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature
and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we
study non-Hopf real hypersurfaces satisfying the equality case of the
inequality under the condition that the mean curvature is constant along each
integral curve of the Reeb vector field. We describe how to obtain all such
hypersurfaces.
