A second order quadratic integral inequality associated with regular problems

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<abstract><p>In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all $ f\in D $,</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray} \int_{a}^{b}{r|f''|^{2}+p|f'|^{2}+q|f|^{2}}\geq\mu_{0}\int_{a}^{b}|f|^{2}. \end{eqnarray} $\end{document} </tex-math></disp-formula></p> <p>Here $ r, p, q $ are real- valued coefficient functions on the compact interval $ [a, b] $ with $ r(x) > 0 $. $ D $ is a linear manifold in the Hilbert function space $ L^{2}(a, b) $ such that all integrals of the above inequality are finite and $ \mu_{0} $ is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in $ L^{2}(a, b) $. The inequality is the best possible, i.e., the number $ \mu_{0} $ cannot be increased. $ f $ is a complex-valued function in $ D. $</p></abstract>