Proving the complete monotonicity of $f$ given $(-log f)'$ is completely monotone
$begingroup$
A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$.
Prove that if $(-log f(x))'$ is completely monotonic, then $f(x)$ is also completely monotonic.
There are a few papers that use this without explicitly displaying the proof. It may help to display the function as $g(x)=(-log f(x))'$ then put $f(x)=exp(-g(x))$ but I have not yet find the pattern to induct the theorem.
analysis monotone-functions
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
$endgroup$
add a comment |
$begingroup$
A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$.
Prove that if $(-log f(x))'$ is completely monotonic, then $f(x)$ is also completely monotonic.
There are a few papers that use this without explicitly displaying the proof. It may help to display the function as $g(x)=(-log f(x))'$ then put $f(x)=exp(-g(x))$ but I have not yet find the pattern to induct the theorem.
analysis monotone-functions
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
$endgroup$
add a comment |
$begingroup$
A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$.
Prove that if $(-log f(x))'$ is completely monotonic, then $f(x)$ is also completely monotonic.
There are a few papers that use this without explicitly displaying the proof. It may help to display the function as $g(x)=(-log f(x))'$ then put $f(x)=exp(-g(x))$ but I have not yet find the pattern to induct the theorem.
analysis monotone-functions
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
$endgroup$
A function $f:(0,∞)→[0,∞]$ is said to be completely monotonic if its $n$-th derivative exists and $(−1)^nf^{(n)}(x)≥0$, where $f^{(n)}(x)$ is the $n$-th derivative of $f$.
Prove that if $(-log f(x))'$ is completely monotonic, then $f(x)$ is also completely monotonic.
There are a few papers that use this without explicitly displaying the proof. It may help to display the function as $g(x)=(-log f(x))'$ then put $f(x)=exp(-g(x))$ but I have not yet find the pattern to induct the theorem.
analysis monotone-functions
analysis monotone-functions
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
edited Dec 18 '18 at 10:34
adli farhan
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
asked Dec 18 '18 at 7:42
adli farhanadli farhan
214
214
add a comment |
add a comment |
1 Answer
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$begingroup$
For an open (and not necessarily connected) set $Asubseteqmathbb{R}$, let $P^infty(A)$ be the set of nonnegative infinitely differentiable functions $Atomathbb{R}$ with all the derivatives also nonnegative.
Now if $g : Ato B$ with $A,Bsubseteqmathbb{R}$ open, $g'in P^infty(A)$, and $hin P^infty(B)$, then $hcirc gin P^infty(A)$ (this can be proven immediately or just seen from this formula).
Your claim reduces to this (consider $g(x)=log f(-x)$ and $h(x)=e^x$).
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
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1 Answer
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1 Answer
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votes
$begingroup$
For an open (and not necessarily connected) set $Asubseteqmathbb{R}$, let $P^infty(A)$ be the set of nonnegative infinitely differentiable functions $Atomathbb{R}$ with all the derivatives also nonnegative.
Now if $g : Ato B$ with $A,Bsubseteqmathbb{R}$ open, $g'in P^infty(A)$, and $hin P^infty(B)$, then $hcirc gin P^infty(A)$ (this can be proven immediately or just seen from this formula).
Your claim reduces to this (consider $g(x)=log f(-x)$ and $h(x)=e^x$).
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
$endgroup$
add a comment |
$begingroup$
For an open (and not necessarily connected) set $Asubseteqmathbb{R}$, let $P^infty(A)$ be the set of nonnegative infinitely differentiable functions $Atomathbb{R}$ with all the derivatives also nonnegative.
Now if $g : Ato B$ with $A,Bsubseteqmathbb{R}$ open, $g'in P^infty(A)$, and $hin P^infty(B)$, then $hcirc gin P^infty(A)$ (this can be proven immediately or just seen from this formula).
Your claim reduces to this (consider $g(x)=log f(-x)$ and $h(x)=e^x$).
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
$endgroup$
add a comment |
$begingroup$
For an open (and not necessarily connected) set $Asubseteqmathbb{R}$, let $P^infty(A)$ be the set of nonnegative infinitely differentiable functions $Atomathbb{R}$ with all the derivatives also nonnegative.
Now if $g : Ato B$ with $A,Bsubseteqmathbb{R}$ open, $g'in P^infty(A)$, and $hin P^infty(B)$, then $hcirc gin P^infty(A)$ (this can be proven immediately or just seen from this formula).
Your claim reduces to this (consider $g(x)=log f(-x)$ and $h(x)=e^x$).
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
$endgroup$
For an open (and not necessarily connected) set $Asubseteqmathbb{R}$, let $P^infty(A)$ be the set of nonnegative infinitely differentiable functions $Atomathbb{R}$ with all the derivatives also nonnegative.
Now if $g : Ato B$ with $A,Bsubseteqmathbb{R}$ open, $g'in P^infty(A)$, and $hin P^infty(B)$, then $hcirc gin P^infty(A)$ (this can be proven immediately or just seen from this formula).
Your claim reduces to this (consider $g(x)=log f(-x)$ and $h(x)=e^x$).
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
edited Dec 18 '18 at 10:10
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
answered Dec 18 '18 at 9:36
metamorphymetamorphy
3,6921621
3,6921621
add a comment |
add a comment |
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