Difference between revisions of "Delta Wirtinger inequality"
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| − | + | ==Theorem== | |
| − | + | Let $M$ be positive and strictly monotone such that $M^{\Delta}$ exists an is [[continuity | rd-continuous]]. Then we have | |
$$\displaystyle\int_a^b |M^{\Delta}(t)|(y^{\sigma}(t))^2 \Delta t \leq \Psi \displaystyle\int_a^b \dfrac{M(t)M^{\sigma}(t)}{|M^{\Delta}(t)|} (y^{\Delta}(t))^2 \Delta t$$ | $$\displaystyle\int_a^b |M^{\Delta}(t)|(y^{\sigma}(t))^2 \Delta t \leq \Psi \displaystyle\int_a^b \dfrac{M(t)M^{\sigma}(t)}{|M^{\Delta}(t)|} (y^{\Delta}(t))^2 \Delta t$$ | ||
for any $y$ with $y(a)=y(b)=0$ and such that $y^{\Delta}$ exists and is rd-continuous, where | for any $y$ with $y(a)=y(b)=0$ and such that $y^{\Delta}$ exists and is rd-continuous, where | ||
$$\Psi = \left\{ \left( \sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right)^{\frac{1}{2}} + \left[\left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{\mu(t)|M^{\Delta}(t)|}{M^{\sigma}(t)} \right) + \left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right) \right]^{\frac{1}{2}} \right\}^2.$$ | $$\Psi = \left\{ \left( \sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right)^{\frac{1}{2}} + \left[\left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{\mu(t)|M^{\Delta}(t)|}{M^{\sigma}(t)} \right) + \left(\sup_{t \in [a,b] \cap \mathbb{T}} \dfrac{M(t)}{M^{\sigma}(t)} \right) \right]^{\frac{1}{2}} \right\}^2.$$ | ||
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| − | + | ==Proof== | |
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==References== | ==References== | ||
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{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 00:39, 15 September 2016
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
References
R. Agarwal, M. Bohner, A. Peterson - Inequalities on Time Scales: A Survey
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
