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first eigenvalue; Sturm-Liouville problem; weight integral condition
Let $\lambda _1(Q)$ be the first eigenvalue of the Sturm-Liouville problem $$ y''-Q(x)y+\lambda y=0,\quad y(0)=y(1)=0,\quad 0<x<1. $$ 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 $\int _0^1x^\alpha (1-x)^\beta Q^{\gamma }(x) {\rm d} x=1$ $(\alpha ,\beta ,\gamma \in \mathbb {R},\gamma \neq 0)$.
[1] Egorov, Yu. V., Kondrat'ev, V. A.: Estimates for the first eigenvalue in some Sturm-Liouville problems. Russ. Math. Surv. 51 (1996), translation from Usp. Math. Nauk 51 (1996), 73-144. MR 1406051 | Zbl 0883.34027
[2] Kuralbaeva, K. Z.: On estimate of the first eigenvalue of a Sturm-Liouville operator. Differents. Uravn. 32 852-853 (1996).
[3] Besov, O. V., Il'in, V. P., Nikol'skiy, S. M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moskva (1996), Russian. MR 1450401
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