We derive Lyapunov-type inequalities for general third-order nonlinear equations involving multiple ψ-Laplacian operators of the form

$$(\psi_{2}((\psi_{1}(u’))’))’ + q(x)f(u) = 0,$$

where

$$\psi_{2}\text{ and }\psi_{1}$$ are odd, increasing functions, ψ1 is sub-multiplicative and 1/ψ1 is convex, and f is a continuous function which satisfies a sign condition. Our results utilize q+ and q-, as opposed to |q| which appears in most results in the literature. Additionally, these new inequalities cover previously obtained results and notably do not require the use of Holder’s or the Cauchy-Schwartz inequality, unlike other proofs. Furthermore, using the obtained inequalities, we obtain a lower bound for the number of zeroes and properties of oscillatory solutions. We conclude with a constraint on the location of the maximum of a solution.

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