Explain to me the concept of global bounds in Jensen's Inequality
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I'm trying to understand some journals on Jensen's Inequality, and I noticed that the term "global bounds" is used often. I can't distinguish this from the "usual" bounds I encountered in my analysis courses. I also literally can't find an article explaining global bounds on the internet, so I figured this may be a jargon among mathematicians who work with inequalities. What is it? How is it different from a "local bound" if such a thing exists?
functional-analysis probability-theory inequality
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
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add a comment |
$begingroup$
I'm trying to understand some journals on Jensen's Inequality, and I noticed that the term "global bounds" is used often. I can't distinguish this from the "usual" bounds I encountered in my analysis courses. I also literally can't find an article explaining global bounds on the internet, so I figured this may be a jargon among mathematicians who work with inequalities. What is it? How is it different from a "local bound" if such a thing exists?
functional-analysis probability-theory inequality
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
$endgroup$
2
$begingroup$
A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03
add a comment |
$begingroup$
I'm trying to understand some journals on Jensen's Inequality, and I noticed that the term "global bounds" is used often. I can't distinguish this from the "usual" bounds I encountered in my analysis courses. I also literally can't find an article explaining global bounds on the internet, so I figured this may be a jargon among mathematicians who work with inequalities. What is it? How is it different from a "local bound" if such a thing exists?
functional-analysis probability-theory inequality
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
$endgroup$
I'm trying to understand some journals on Jensen's Inequality, and I noticed that the term "global bounds" is used often. I can't distinguish this from the "usual" bounds I encountered in my analysis courses. I also literally can't find an article explaining global bounds on the internet, so I figured this may be a jargon among mathematicians who work with inequalities. What is it? How is it different from a "local bound" if such a thing exists?
functional-analysis probability-theory inequality
functional-analysis probability-theory inequality
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
asked Jan 25 at 15:50
Liam CeasarLiam Ceasar
586
586
2
$begingroup$
A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03
add a comment |
2
$begingroup$
A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03
2
2
$begingroup$
A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03
$begingroup$
A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03
add a comment |
1 Answer
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Yes, Jensens inequality states for a convex function $f:[a,b] to mathbb{R}$, and a finite sequence $p_i in [0,1] $ with $sum p_i = 1$ and a finite sequence $x_iin[a,b]$ it holds
$$ 0 le sum_i {p_i} f(x_i) - f( sum_i p_i x_i) $$
And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"
E.g a well known result in case $f$ is additionally differentable there is an upper bound:
$$ sum_i {p_i} f(x_i) - f( sum_i p_i x_i) le frac{1}{4}(b-a)(f'(b)-f'(a))$$
So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'.
If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.
However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.
answered Jan 25 at 17:51
MaksimMaksim
78718
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add a comment |
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$begingroup$
Yes, Jensens inequality states for a convex function $f:[a,b] to mathbb{R}$, and a finite sequence $p_i in [0,1] $ with $sum p_i = 1$ and a finite sequence $x_iin[a,b]$ it holds
$$ 0 le sum_i {p_i} f(x_i) - f( sum_i p_i x_i) $$
And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"
E.g a well known result in case $f$ is additionally differentable there is an upper bound:
$$ sum_i {p_i} f(x_i) - f( sum_i p_i x_i) le frac{1}{4}(b-a)(f'(b)-f'(a))$$
So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'.
If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.
However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.
answered Jan 25 at 17:51
MaksimMaksim
78718
$endgroup$
add a comment |
$begingroup$
Yes, Jensens inequality states for a convex function $f:[a,b] to mathbb{R}$, and a finite sequence $p_i in [0,1] $ with $sum p_i = 1$ and a finite sequence $x_iin[a,b]$ it holds
$$ 0 le sum_i {p_i} f(x_i) - f( sum_i p_i x_i) $$
And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"
E.g a well known result in case $f$ is additionally differentable there is an upper bound:
$$ sum_i {p_i} f(x_i) - f( sum_i p_i x_i) le frac{1}{4}(b-a)(f'(b)-f'(a))$$
So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'.
If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.
However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.
answered Jan 25 at 17:51
MaksimMaksim
78718
$endgroup$
add a comment |
$begingroup$
Yes, Jensens inequality states for a convex function $f:[a,b] to mathbb{R}$, and a finite sequence $p_i in [0,1] $ with $sum p_i = 1$ and a finite sequence $x_iin[a,b]$ it holds
$$ 0 le sum_i {p_i} f(x_i) - f( sum_i p_i x_i) $$
And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"
E.g a well known result in case $f$ is additionally differentable there is an upper bound:
$$ sum_i {p_i} f(x_i) - f( sum_i p_i x_i) le frac{1}{4}(b-a)(f'(b)-f'(a))$$
So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'.
If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.
However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.
answered Jan 25 at 17:51
MaksimMaksim
78718
$endgroup$
Yes, Jensens inequality states for a convex function $f:[a,b] to mathbb{R}$, and a finite sequence $p_i in [0,1] $ with $sum p_i = 1$ and a finite sequence $x_iin[a,b]$ it holds
$$ 0 le sum_i {p_i} f(x_i) - f( sum_i p_i x_i) $$
And now you could be tempted to ask how big the right hand side of this inequality might get - or in other words : "is there an upper bound for the Jensen's inequality?"
E.g a well known result in case $f$ is additionally differentable there is an upper bound:
$$ sum_i {p_i} f(x_i) - f( sum_i p_i x_i) le frac{1}{4}(b-a)(f'(b)-f'(a))$$
So we have an upper bound which does not involve the $x_i$ or $p_i$, which is oftenly considered as a 'global bound'.
If it involves $x_i$ or $p_i$, then people talk about a 'local bound'.
However the terms 'global bound' and 'local bound' are not strictly defined under all authors publishing papers on this topic.
answered Jan 25 at 17:51
MaksimMaksim
78718
edited Jan 28 at 21:47
answered Jan 25 at 17:51
MaksimMaksim
78718
answered Jan 25 at 17:51
MaksimMaksim
78718
answered Jan 25 at 17:51
MaksimMaksim
78718
78718
add a comment |
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A global bound would hold everywhere in the domain of the function, and a local bound only in a neighborhood of some point, I would say. It would be easier to be sure if you would give an example of the use of the phrase.
$endgroup$
– saulspatz
Jan 25 at 16:03