Difference between revisions of "Delta Jensen inequality"
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| − | + | __NOTOC__ | |
| − | + | ==Theorem== | |
| + | Let $a,b \in \mathbb{T}$ and $c,d \in \mathbb{R}$. Suppose $g \colon [a,b]\cap \mathbb{T} \rightarrow (c,d)$ is [[rd-continuous]] and $F \colon (c,d) \rightarrow \mathbb{R}$ is convex. Then | ||
$$F \left(\dfrac{\displaystyle\int_a^b g(t) \Delta t}{b-a}\right) \leq \dfrac{\displaystyle\int_a^b F(g(t))\Delta t}{b-a}.$$ | $$F \left(\dfrac{\displaystyle\int_a^b g(t) \Delta t}{b-a}\right) \leq \dfrac{\displaystyle\int_a^b F(g(t))\Delta t}{b-a}.$$ | ||
| − | + | ||
| − | + | ==Proof== | |
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==References== | ==References== | ||
| − | + | {{PaperReference|Inequalities on Time Scales: A Survey|2001|Ravi Agarwal|author2 = Martin Bohner| author3 = Allan Peterson|prev=findme|next=findme}}: Theorem 4.1 | |
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| + | {{:Delta inequalities footer}} | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 00:36, 15 September 2016
Theorem
Let $a,b \in \mathbb{T}$ and $c,d \in \mathbb{R}$. Suppose $g \colon [a,b]\cap \mathbb{T} \rightarrow (c,d)$ is rd-continuous and $F \colon (c,d) \rightarrow \mathbb{R}$ is convex. Then $$F \left(\dfrac{\displaystyle\int_a^b g(t) \Delta t}{b-a}\right) \leq \dfrac{\displaystyle\int_a^b F(g(t))\Delta t}{b-a}.$$
Proof
References
Ravi Agarwal, Martin Bohner and Allan Peterson: Inequalities on Time Scales: A Survey (2001)... (previous)... (next): Theorem 4.1
$\Delta$-Inequalities
| Bernoulli | Bihari | Cauchy-Schwarz | Gronwall | Hölder | Jensen | Lyapunov | Markov | Minkowski | Opial | Tschebycheff | Wirtinger |
