A note on inequalities related to Opial's inequality (Q1891540)

The author establishes some new generalizations of the Opial integral inequality involving an unknown function of one variable and its higher order derivatives. Let \(n\geq 1\) be a fixed integer, \(f\in \mathbb{C}^n(I, \mathbb{R})\), \(I:= [a,b]\), \(R:=(- \infty, \infty)\), \(w\) be a positive weight function on \(I\), \(v\) another weight function, \(p\geq 1\), \(q> 0\) and \(r_k\geq 0\) \((k= 0, 1,\dots, n- 1)\) be real numbers with \(r_0+ r_1+\cdots+ r_{n- 1}= 1\). In the Theorem 1, it is proved that if the condition \[ f^{(k)}(a)= 0\qquad\text{for}\qquad k= 0, 1,\dots, n- 1\tag{C} \] is satisfied, then \[ \int^b_a w \Biggl[ \prod^{n- 1}_{k= 0} |f^{(k)}|^{r_k}\Biggr]^p |f^{(n)}|^q dx\leq Q_1(p, q) \int^b_a v|f^{(n)}|^{p+ q} dx,\tag{I} \] where \(Q_1\) is a suitable positive constant defined in the note that is too long to be cited here. In Theorem 2, it is proved that if condition (C) is replaced by \(f^{(k)}(b)= 0\) \((k= 0, 1, \dots, n- 1)\) then (I) holds with another positive constant \(Q_2(p, q)\). These results improve some results of \textit{W. S. Cheung} [Mathematika 37, No. 1, 136-142 (1990; Zbl 0706.26015)]. Some special cases of these results are slight variants of some known inequalities due to \textit{P. R. Beesack} and \textit{K. M. Das} [Pac. J. Math. 26, 215-232 (1968; Zbl 0162.079)], \textit{K. M. Das} [Proc. Am. Math. Soc. 22, 258-261 (1969; Zbl 0185.123)], and \textit{G. Yang} [Proc. Jap. Acad. 42, 78-83 (1966; Zbl 0151.052)]. There are some misprints in the paper.