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 $t,s \in \mathbb{T}$
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Let $\mathbb{T}$ be a [[time scale]] and $\alpha \in \mathbb{R}$ be a [[Regressive_function | positively regressive]] constant. Then for all $t,s \in \mathbb{T}$
$$e_{\alpha} \geq 1 + \alpha(t-s).$$
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$$e_{\alpha} \geq 1 + \alpha(t-s),$$
 +
where $e_{\alpha}$ denotes the [[delta exponential]].
  
 
==Proof==
 
==Proof==

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 AgarwalMartin 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