On determination of best-possible constants in integral inequalities involving derivatives

Abstract:

This paper is concerned with the numerical approximation of the best possible constants ${\gamma _{n,k}}$ in the inequality \[ {\left \| {{F^{(k)}}} \right \|^2} \leqslant \gamma _{n,k}^{ - 1}\;\left \{ {{{\left \| F \right \|}^2} + {{\left \| {{F^{(n)}}} \right \|}^2}} \right \},\] where \[ {\left \| F \right \|^2} = \int _0^\infty |F(x){|^2}\;dx.\] A list of all constants ${\gamma _{n,k}}$ for $n \leqslant 10$ is given.