Opial-type inequalities involving higher order derivatives
In the present paper, some new Opial-type integral inequalities are obtained, which generalize many known results due to the first author and \textit{E. Thandapani} [An. Ştiinţ. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 28, 123-126 (1982; Zbl 0556.26007)], \textit{W.-S. Cheung} [Mathematika 37, No. 1, 136-142 (1990; Zbl 0706.26015)], \textit{K. M. Das} [Proc. Am. Math. Soc. 22, 258-261 (1969; Zbl 0185.123)], \textit{B. G. Pachpatte} [J. Math. Anal. Appl. 120, 547-556 (1986; Zbl 0608.26009)], and \textit{G.-S. Yang} [Tamkang J. Math. 18, No. 4, 101-104 (1987; Zbl 0667.26011)]. Let \(x(t)\in C^{(n- 1)}[\alpha, T]\) be such that \(x^{(i)}(\alpha)= 0\), \(0\leq k\leq i\leq n- 1\) \((n\geq 1)\), \(x^{(n- 1)}(t)\) absolutely continuous on \((\alpha, T)\), and \(x^{(n)}(t)\) does not change sign on \((\alpha, T)\). Let \(p(t)\) and \(q(t)\) be nonnegative and measurable functions defined on \((\alpha, T)\). The main results of the paper are inequalities of the following form \[ \int^ T_ \alpha G_ i[x(s)] ds\leq K_ i\Biggl\{ \int^ T_ \alpha p(s)| x^{(n)}(s)|^ r \Biggr\}^{m_ i},\quad i= 1,2,3,\tag{A\(_ i\)} \] where \[ G_ 1[x(t)]= q(t)| x^{(k)}(t)| ^{r_ k} | x^{(n)}(t)|^{r_ n},\;G_ 2[x(t)]= q(t)\prod^ n_{k= 0} | x^{(k)}(t)|^{r_ k}, \] \[ G_ 3[x(t)]= q(t)| x^{(k)}(t) |^{r_ k} | x^{(k+ 1)}(t)|^{r_{k+1}}, \] and \(m_ 1= (r_ k+ r_ n)/r\), \(m_ 2= (r_ 1+\cdots+ r_ n)/ r\), \(m_ 3= (r_ k+ r_{k+ 1})/r\); \(r> 1\), \(r_ 0,\dots, r_ n\) are real numbers subject to some additional conditions, and \(K_ i\) are some nonnegative real constants depending on \(p,q,r_ 0,\dots, r_ n\), \(r\), \(\alpha\), and \(T\). \{Reviewer's remark: (i) To derive (2.6) from (2.5), we need the positivity (instead of the nonnegativity) of the function \(p\); (ii) The inverse inequalities of \((\text{A}_ i)\) given herein should be reproved if possible. Because, with \(T\) \((T\geq t)\) as the upper limit of integration, the first equality sign of (2.1) ought to be replaced by the inequality sign ``\(\leq\)''\}.
scientific article; zbMATH DE number 743887