Difference between revisions of "Delta Bernoulli inequality"
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==Theorem== | ==Theorem== | ||
| − | Let $\alpha \in \mathbb{R}$ be a [[Regressive_function | positively regressive]] constant. Then for all $t,s \in \mathbb{T}$ | + | Let $\mathbb{T}$ be a [[time scale]] and $\alpha \in \mathbb{R}$ be a [[Regressive_function | positively regressive]] constant. Then for all $t,s \in \mathbb{T}$ |
| − | $$e_{\alpha} \geq 1 + \alpha(t-s) | + | $$e_{\alpha} \geq 1 + \alpha(t-s),$$ |
| + | where $e_{\alpha}$ denotes the [[delta exponential]]. | ||
==Proof== | ==Proof== | ||
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{{:Delta inequalities footer}} | {{:Delta inequalities footer}} | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 15:45, 21 January 2023
Theorem
Let $\mathbb{T}$ be a time scale and $\alpha \in \mathbb{R}$ be a positively regressive constant. Then for all $t,s \in \mathbb{T}$ $$e_{\alpha} \geq 1 + \alpha(t-s),$$ where $e_{\alpha}$ denotes the delta exponential.
Proof
References
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 5.5
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
