General stability criterion of inviscid parallel flow

A more restrictively general stability criterion of two-dimensional inviscid parallel flow is obtained analytically. First, a sufficient criterion for stability is found as either $-\mu_1<\frac{U''}{U-U_s}<0$ or $0<\frac{U''}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection point, $\mu_1$ is the eigenvalue of Poincar\'{e}'s problem. Second, this criterion is generalized to barotropic geophysical flows in $\beta$ plane. Based on the criteria, the flows are are divided into different categories of stable flows, which may simplify the further investigations. And the connections between present criteria and Arnol'd's nonlinear criteria are discussed. These results extend the former criteria obtained by Rayleigh, Tollmien and Fj{\o}rtoft and would intrigue future research on the mechanism of hydrodynamic instability.