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Lyapunov and Wirtinger inequalities. (English) Zbl 1062.34005
Summary: We prove the Lyapunov inequality for the second-order linear differential equation $$\biggl(r(t)\varphi\bigl(y'(t)\bigr) \biggr)'+ p(t) \varphi\bigl(y(t)\bigr)=0,$$ where (i) $\varphi(s)=|s|^{\alpha-2}s$, $\alpha >1$ is a fixed real number, (ii) $r(t)$ and $p(t)$ are integrable on $[a,b]$ with $r(t)>0$ on $[a,b]$. On the other hand, a generalized Wirtinger inequality is also given.

34A30Linear ODE and systems, general
26D15Inequalities for sums, series and integrals of real functions
Full Text: DOI
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