Maria Telnova, Moscow State University of Economics, Statistics, and Informatics, 119501, Nezhinskaya st., 7, Moskva, Russia, e-mail: mytelnova@ya.ru
Abstract: 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,\quad0<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 $\int_0^1x^\alpha(1-x)^\beta Q^{\gamma}(x) d x=1$ $(\alpha,\beta,\gamma\in\mathbb{R},\gamma\neq0)$.
Keywords: first eigenvalue, Sturm-Liouville problem, weight integral condition
Classification (MSC 2010): 34L15
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