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Published online by Cambridge University Press: 14 February 2012
Conjugate points are defined in terms of solutions of a linear fourth-order differential equation satisfying two homogeneous boundary conditions at x = α and either u(β) = u′(β) = 0 or u′(γ) = u″(γ) = 0. The smallest β > α and γ > α such that these boundary conditions are satisfied by a non-trivial solution of the equation are denoted by η(α) and ῆ-(α), respectively. Upper bounds are established for min [η(α), ῆ(α)] relative to the conjugate points of a self-adjoint differential equation which is majorised by the more general equation under study.