Let λ1(Q) be the first eigenvalue of the Sturm-Liouville problem y ′′ − Q(x)y + λy = 0, y(0) = y(1) = 0, 0 < x < 1. We give some estimates for mα,β,γ = inf Q∈Tα,β,γ λ1(Q) and Mα,β,γ = sup Q∈Tα,β,γ λ1(Q), where Tα,β,γ is the set of real-valued measurable on [0, 1] x α(1 − x) β -weighted Lγ-functions Q with non-negative values such that ∫ 1 0 x α(1 − x) βQ γ (x) dx = 1 (α, β, γ ∈ ℝ, γ ≠ 0).
We consider the Sturm-Liouville problem with symmetric boundary conditions and an integral condition. We estimate the first eigenvalue λ1 of this problem for different values of the parameters.