Sharp Integral Inequalities Involving High-Order Partial Derivatives

Inequalities involving functions of n independent variables, their partial derivatives, integrals play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations 1–10 . Especially, in view of wider applications, inequalities due to Agarwal, Opial, Pachpatte, Wirtinger, Poincaré and et al. have been generalized and sharpened from the very day of their discover. As a matter of fact, these now have become research topic in their own right 11–14 . In the present paper, we will use the same method of Agarwal and Sheng 15 , establish some new estimates on these types of inequalities involving higher-order partial derivatives. We further generalize these inequalities which lead to result sharper than those currently available. An important characteristic of our results is that the constant in the inequalities are explicit.