Difference between revisions of "Delta Hölder inequality"

From timescalewiki
Jump to: navigation, search
Line 8: Line 8:
  
 
==References==
 
==References==
{{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 3.1
+
{{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 3.2
  
 
{{:Delta inequalities footer}}
 
{{:Delta inequalities footer}}

Revision as of 00:27, 15 September 2016

Theorem

Let $a,b \in \mathbb{T}$. For rd-continuous $f,g \colon [a,b]\cap\mathbb{T} \rightarrow \mathbb{R}$ we have $$\displaystyle\int_a^b |f(t)g(t)| \Delta t \leq \left( \displaystyle\int_a^b |f(t)|^p \Delta t \right)^{\frac{1}{p}} \left(\displaystyle\int_a^b |g(t)|^q \Delta t \right)^{\frac{1}{q}}$$ where $p>1$ and $q = \dfrac{p}{p-1}$.

Proof

References

Ravi AgarwalMartin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 3.2

$\Delta$-Inequalities

Bernoulli Bihari Cauchy-Schwarz Gronwall Hölder Jensen Lyapunov Markov Minkowski Opial Tschebycheff Wirtinger