An extension of the sharp Wirtinger inequality

In this study, we give a sharp Wirtinger inequality \left\| f \right\|_p \text≤ C(p\rm,q)(b - a)^r + 1 / p - 1 / q \left\| f^(r) \right\|_q\rm,\;\; 1 \text≤ p\rm, q \text≤ \infty . For an arbitrary f \in W_q^ra\rm,b with f(x_1) = f(x_2) = \cdots = f(x_r) = 0,a \text≤ x_1 \text＜ x_2\text＜ \cdots \text＜ x_r \text≤ b．From integral type remainder of Lagrange interpolation, we refer computation of C(p\rm,q) to the norm of an integral operator. We refer values of C(1\rm,1) and C(\infty \rm,\infty ) to two explicit integral expressions and value of C(2\rm,2) to computation of maximum eigenvalue of a Hilbert-Schmidt operator．An example was then given to show our method．