On a Class of Integral Inequalities

The Titchmarsh‐Weyl coefficient for the differential equation −( x β y ′ ( x ) ) ′ = λ x α y ( x )   ( x ∈ ( 0 , ∞ ) )where the real parameters α and β satisfy α = − 1 and β < 1, is calculated explicitly. The result is applied to obtain the best possible constant K(α,β) in the integral inequality(∫ 0 ∞x β f ′( x ) 2 d x) ⩽ K ( α , β )∫ 0 ∞x α f ( x )2 d x∫ 0 ∞x − α{( x β f ′ ( x ) ) ′ } 2 d x .