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Sharp Opial-type inequalities involving higher order derivatives of two functions. (English) Zbl 0831.26007

Let \(p(t)\) and \(q(t)\) be non-negative, measurable functions in \([\alpha, T]\), \(-\infty\leq \alpha\leq T< \infty\). Let \(n\in \mathbb{N}\), \(0\leq k\leq n- 1\) be fixed and let \(r_n> 0\), \(r_k\geq 0\), \(r> \max\{1, r_n\}\) be given numbers. Suppose that \(x_1(t),x_2(t)\in C^{(n- 1)}[\alpha, T]\) satisfy \(x_1^{(n- 1)}(t), x_2^{(n- 1)}(t)\in AC[\alpha, T]\) and \(x_1^{(i)}(\alpha)= x^{(i)}_2(\alpha)= 0\) for \(k\leq i\leq n- 1\). Then \[ \begin{split} \int^T_\alpha q(t) \Biggl(\sum^2_{j= 1} |x^{(k)}_j(t)|^{r_k} |x_j^{(n)}(t)|^{r_n}\Biggr)dt\leq\\ K_1(p, q, r_k, r_n, r) \Biggl[\int^T_\alpha p(t) \Biggl( \sum^2_{j= 1} |x^{(n)}_j(t)|\Biggr) dt\Biggr]^{(r_k+ r_n)/r}.\end{split} \] This is one of the main results of the paper, the others are its modifications.
Reviewer: B.Opic (Praha)

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
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References:

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