On Opial type inequalities involving higher order derivatives (Q1892240)

The author establishes three new Opial-type integral inequalities involving a function and its higher order derivatives and the highest order derivative appearing in both of its sides. The main result of the paper is embodied in Theorem 1. It shows that \[ \int^b_a w|f^{(i)}|^{p_1} |f^{(j)}|^{p_2} |f^{(n)}|^{p_3} dt\leq \Biggl({p_3\over p}\Biggr)^{p_3/p} M_1 \int^b_a v|f^{(n)}|^p dt\tag{1} \] is valid, where \(p_1\), \(p_2\), \(p_3\) are nonnegative real numbers, \(p= p_1+ p_2+ p_3> p_3> 0\) and \(p> 1\), \(0\leq i\leq j\leq n- 1\), \(n\geq 2\), \(w\), \(v\) are weight functions defined on \(I:= [a, b]\), and \(f\in C^n(I, \mathbb{R})\) satisfying \(f(a)= f'(a)=\cdots= f^{(n- 1)}(a)= 0\), and \(M_1\) is a suitable positive number defined in the paper. The results are proved by elementary methods and which add a new range of Opial-type inequalities. The uniqueness of the Cauchy-type problem \[ [r(t) y^{(n)}]^{(n+ 1)}- g(t) y(t)= h(t),\qquad t\in [a, b] \] with \[ y^{(i- 1)}(a)= 0,\quad [r(a) y^{(n)}(a)]^{(i)}= 0,\qquad i= 1, 2,\dots, n \] is considered to convey usefulness of the results obtained.