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?







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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).
    – 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.
    – Adam
    Aug 29 at 11:05







  • 1




    Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
    – gerw
    Aug 29 at 11:30










  • Can you provide some references? Thanks
    – Adam
    Aug 29 at 11:42










  • At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
    – gerw
    Aug 29 at 11:47














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?







share|cite|improve this question




















  • If $L$ maps $mathbb R$ to $mathbb R$ (and if $L$) is continuous, this should be equivalent to monotonicity (decreasing or increasing).
    – 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.
    – Adam
    Aug 29 at 11:05







  • 1




    Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
    – gerw
    Aug 29 at 11:30










  • Can you provide some references? Thanks
    – Adam
    Aug 29 at 11:42










  • At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
    – gerw
    Aug 29 at 11:47












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?







share|cite|improve this question












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?









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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).
    – 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.
    – Adam
    Aug 29 at 11:05







  • 1




    Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
    – gerw
    Aug 29 at 11:30










  • Can you provide some references? Thanks
    – Adam
    Aug 29 at 11:42










  • At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
    – 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).
    – 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.
    – Adam
    Aug 29 at 11:05







  • 1




    Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
    – gerw
    Aug 29 at 11:30










  • Can you provide some references? Thanks
    – Adam
    Aug 29 at 11:42










  • At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
    – 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).
– 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).
– 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.
– 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.
– Adam
Aug 29 at 11:05





1




1




Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
– gerw
Aug 29 at 11:30




Your functional is quasiconvex and quasiconcave. Sometimes this is called 'quasilinear'.
– gerw
Aug 29 at 11:30












Can you provide some references? Thanks
– Adam
Aug 29 at 11:42




Can you provide some references? Thanks
– Adam
Aug 29 at 11:42












At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
– gerw
Aug 29 at 11:47




At least it's on wikipedia: en.wikipedia.org/wiki/Quasiconvex_function
– gerw
Aug 29 at 11:47















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