Difference between revisions of "Delta Wirtinger inequality"

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==Theorem==
<strong>Theorem:</strong> Let $M$ be positive and strictly monotone such that $M^{\Delta}$ exists an is [[continuity | rd-continuous]]. Then we have
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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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<strong>Proof:</strong> █
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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]]
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[[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