On certain integral inequalities related to Opial's inequality (Q1084200)

The aim of the present paper is to establish some new integral inequalities involving three functions and their derivatives which in special cases yield the well known Opial inequality and some of its generalizations. One of the results reads as follows. Let f,g,h be absolutely continuous functions on \([a,b]\) which vanish at a and b. Let \(m\geq 0\) be a real number. Denote \(K(f,g)=| fg|^ m(| fg'| +| gf'|).\) Then: \[ \int^{b}_{a}(K(f,g)+K(g,h)+K(h,f))dx\leq (1/(m+1))((b- a)/2)^{2m+1}\int^{b}_{a}(| f'|^ 2+| g'|^ 2+| h'|^ 2)dx. \] The equality case is also discussed. Opial's inequality is obtained when \(a=0,\) \(m=0\) and \(h=g=f\).