On Lyapunov-type inequalities for fourth order linear differential equations

In this  paper,  we introduce  new  Lyapunov-type inequalities  for fourth order  linear  di¤erential  equations  under  the  third-point  and  four-point  boundary conditions.  The result for four-point boundary conditions improve some existing ones in literature and also, we note that the results for three-point boundary conditions are new.