Functional property name?
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I would like to know if this functional property has a name.
Definition. We say that a functional $L$ has a property if and only if $L(f) leq L(g)$ implies
$$
L(f) leq Lleft(fracf+g2right) leq L(g).
$$
Do you know if this property has a name?
functional-analysis inequality universal-property
add a comment |Â
up vote
1
down vote
favorite
I would like to know if this functional property has a name.
Definition. We say that a functional $L$ has a property if and only if $L(f) leq L(g)$ implies
$$
L(f) leq Lleft(fracf+g2right) leq L(g).
$$
Do you know if this property has a name?
functional-analysis inequality universal-property
If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
1
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I would like to know if this functional property has a name.
Definition. We say that a functional $L$ has a property if and only if $L(f) leq L(g)$ implies
$$
L(f) leq Lleft(fracf+g2right) leq L(g).
$$
Do you know if this property has a name?
functional-analysis inequality universal-property
I would like to know if this functional property has a name.
Definition. We say that a functional $L$ has a property if and only if $L(f) leq L(g)$ implies
$$
L(f) leq Lleft(fracf+g2right) leq L(g).
$$
Do you know if this property has a name?
functional-analysis inequality universal-property
asked Aug 29 at 10:51
Adam
385113
385113
If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
1
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47
add a comment |Â
If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
1
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47
If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
1
1
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47
add a comment |Â
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If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
â gerw
Aug 29 at 10:58
Yes, but I am specially interested in functionals that represent non-linear integrals. Currently, I am interested in decomposition integrals and there is easy to find functions $f,g$ such that $L(f) leq L(g)$ but the inequality $L(f) leq L((f+g)/2) leq L(g)$ does not hold. As other example, the integrals that are linear have the mentioned property.
â Adam
Aug 29 at 11:05
1
Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
â gerw
Aug 29 at 11:30
Can you provide some references? Thanks
â Adam
Aug 29 at 11:42
At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
â gerw
Aug 29 at 11:47