Delta Wirtinger inequality

Theorem: Let $M$ be positive and strictly monotone such that $M^{\Delta}$ exists an is rd-continuous. Then we have $$\displaystyle\int_a^b |M^{\Delta}(t)|(y^{\sigma}(t))^2 \Delta t \leq \Psi \displaystyle\int_a^b \dfrac{M(t)M^{\sigma}(t)}{|M^{\Delta}(t)|} (y^{\Delta}(t))^2 \Delta t$$ for any $y$ with $y(a)=y(b)=0$ and such that $y^{\Delta}$ exists and is rd-continuous, where $$\Psi = \left\{ \left( \sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right)^{\frac{1}{2}} + \left[\left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{\mu(t)|M^{\Delta}(t)|}{M^{\sigma}(t)} \right) + \left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right) \right]^{\frac{1}{2}} \right\}^2.$$

Proof: █