Delta Lyapunov inequality

Theorem: Let $p \colon \mathbb{T} \rightarrow \mathbb{R}^+$ be positive-valued and [continuity | rd-continuous]. If the Sturm-Liouville dynamic equation $$x^{\Delta^2} + p(t) x^{\sigma} = 0$$ has a nontrivial solution $x$ with $x(a)=x(b)=0$, then the Lyapunov inequality $$\displaystyle\int_a^b p(t) \Delta t \geq \dfrac{b-a}{f(d)}$$ holds, where $f \colon \mathbb{T} \rightarrow \mathbb{R}$ is defined by $$f(t) = (t-a)(b-t)$$ and $d \in \mathbb{T}$ is such that $$\left| \dfrac{a+b}{2} - d \right| = \min \left\{ \left| \dfrac{a+b}{2} -s \right| \colon s \in [a,b] \cap \mathbb{T} \right\}.$$

Proof: █

References

R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey