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Some estimates for the first eigenvalue of the Sturm-Liouville problem with a weight integral condition. (English) Zbl 1265.34313
Summary: Let $$\lambda _1(Q)$$ be the first eigenvalue of the Sturm-Liouville problem $y''-Q(x)y+\lambda y=0,\;y(0)=y(1)=0,\quad 0<x<1.$ We give some estimates for $$m_{\alpha ,\beta ,\gamma }=\inf _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$$ and $$M_{\alpha ,\beta ,\gamma }=\sup _{Q\in T_{\alpha ,\beta ,\gamma }}\lambda _1(Q)$$, where $$T_{\alpha ,\beta ,\gamma }$$ is the set of real-valued measurable on $$\left [0,1\right ]$$ $$x^\alpha (1-x)^\beta$$-weighted $$L_\gamma$$-functions $$Q$$ with non-negative values such that $$\int _0^1x^\alpha (1-x)^\beta Q^{\gamma }(x)\, \text{d} x=1$$ $$(\alpha ,\beta ,\gamma \in \mathbb {R},\gamma \neq 0)$$.

##### MSC:
 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34B24 Sturm-Liouville theory
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