A moment inequality











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Let $chi(s)=int_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$tin[0,1]$, and $f(t)geq0$.



Is it possible to show that



$left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)-left(chi(3)chi(1)-chi(2)^{2}right)^{2}geq0$



Note: I believe that



$chi(0)chi(2)-chi(1)^{2}geq0$



$chi(4)chi(2)-chi(3)^{2}geq0$



follows from the Cauchy-Schwarz inequality.
(correct me if I am wrong about this).










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  • Editing to give a more informative title would be useful.
    – YCor
    Nov 20 at 15:13















up vote
3
down vote

favorite
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Let $chi(s)=int_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$tin[0,1]$, and $f(t)geq0$.



Is it possible to show that



$left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)-left(chi(3)chi(1)-chi(2)^{2}right)^{2}geq0$



Note: I believe that



$chi(0)chi(2)-chi(1)^{2}geq0$



$chi(4)chi(2)-chi(3)^{2}geq0$



follows from the Cauchy-Schwarz inequality.
(correct me if I am wrong about this).










share|cite|improve this question









New contributor




hopeless is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Editing to give a more informative title would be useful.
    – YCor
    Nov 20 at 15:13













up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





Let $chi(s)=int_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$tin[0,1]$, and $f(t)geq0$.



Is it possible to show that



$left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)-left(chi(3)chi(1)-chi(2)^{2}right)^{2}geq0$



Note: I believe that



$chi(0)chi(2)-chi(1)^{2}geq0$



$chi(4)chi(2)-chi(3)^{2}geq0$



follows from the Cauchy-Schwarz inequality.
(correct me if I am wrong about this).










share|cite|improve this question









New contributor




hopeless is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Let $chi(s)=int_{0}^{1}x(t)^{s}f(t)dt$,
where $x(t)$ and $f(t)$ are real valued continuous functions for
$tin[0,1]$, and $f(t)geq0$.



Is it possible to show that



$left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)-left(chi(3)chi(1)-chi(2)^{2}right)^{2}geq0$



Note: I believe that



$chi(0)chi(2)-chi(1)^{2}geq0$



$chi(4)chi(2)-chi(3)^{2}geq0$



follows from the Cauchy-Schwarz inequality.
(correct me if I am wrong about this).







real-analysis inequalities cauchy-schwarz-inequality






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hopeless is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











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edited Nov 20 at 15:14









Iosif Pinelis

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asked Nov 20 at 12:59









hopeless

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184




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hopeless is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Check out our Code of Conduct.












  • Editing to give a more informative title would be useful.
    – YCor
    Nov 20 at 15:13


















  • Editing to give a more informative title would be useful.
    – YCor
    Nov 20 at 15:13
















Editing to give a more informative title would be useful.
– YCor
Nov 20 at 15:13




Editing to give a more informative title would be useful.
– YCor
Nov 20 at 15:13










1 Answer
1






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up vote
6
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accepted










This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)ge R(x)$, where $L(x):=left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)$ and $R(x):=chi(3)chi(1)-chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz inequality, $R(x)>0$. Then for small enough $t>0$ we have $L(tx)=t^8L(x)<t^4R(x)=R(tx)$.





Added in response to the OP fixing the typo: The inequality is now true. Indeed, let $m_j:=chi(j)$. Without loss of generality, $m_0 m_2-m_1^2>0$, by the Cauchy--Schwarz inequality. The matrix $M:=[m_{i+j}]_{i,j=0}^2$ is positive semidefinite. This follows because
begin{equation}
0leint_0^1Big(sum_{i=0}^2 a_i, x(t)^iBig)^2f(t),dt=sum_{i,j=0}^2 m_{i+j}a_i a_j
end{equation}

for all real $a_0,a_1,a_2$.



So, the determinant of $M$ is $ge0$, which is equivalent to
begin{equation}
m_4ge m_4^*:=frac{m_2^3-2 m_1 m_3 m_2+m_0 m_3^2}{m_0 m_2-m_1^2}.
end{equation}

On the other hand, the difference between the left and right sides of your inequality (with $m_j$ in place of $chi(j)$) is increasing in $m_4$, and the value of this difference is $0$ at $m_4=m_4^*$. Thus, your inequality follows.






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  • Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
    – hopeless
    Nov 20 at 13:28











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1 Answer
1






active

oldest

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
6
down vote



accepted










This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)ge R(x)$, where $L(x):=left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)$ and $R(x):=chi(3)chi(1)-chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz inequality, $R(x)>0$. Then for small enough $t>0$ we have $L(tx)=t^8L(x)<t^4R(x)=R(tx)$.





Added in response to the OP fixing the typo: The inequality is now true. Indeed, let $m_j:=chi(j)$. Without loss of generality, $m_0 m_2-m_1^2>0$, by the Cauchy--Schwarz inequality. The matrix $M:=[m_{i+j}]_{i,j=0}^2$ is positive semidefinite. This follows because
begin{equation}
0leint_0^1Big(sum_{i=0}^2 a_i, x(t)^iBig)^2f(t),dt=sum_{i,j=0}^2 m_{i+j}a_i a_j
end{equation}

for all real $a_0,a_1,a_2$.



So, the determinant of $M$ is $ge0$, which is equivalent to
begin{equation}
m_4ge m_4^*:=frac{m_2^3-2 m_1 m_3 m_2+m_0 m_3^2}{m_0 m_2-m_1^2}.
end{equation}

On the other hand, the difference between the left and right sides of your inequality (with $m_j$ in place of $chi(j)$) is increasing in $m_4$, and the value of this difference is $0$ at $m_4=m_4^*$. Thus, your inequality follows.






share|cite|improve this answer























  • Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
    – hopeless
    Nov 20 at 13:28















up vote
6
down vote



accepted










This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)ge R(x)$, where $L(x):=left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)$ and $R(x):=chi(3)chi(1)-chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz inequality, $R(x)>0$. Then for small enough $t>0$ we have $L(tx)=t^8L(x)<t^4R(x)=R(tx)$.





Added in response to the OP fixing the typo: The inequality is now true. Indeed, let $m_j:=chi(j)$. Without loss of generality, $m_0 m_2-m_1^2>0$, by the Cauchy--Schwarz inequality. The matrix $M:=[m_{i+j}]_{i,j=0}^2$ is positive semidefinite. This follows because
begin{equation}
0leint_0^1Big(sum_{i=0}^2 a_i, x(t)^iBig)^2f(t),dt=sum_{i,j=0}^2 m_{i+j}a_i a_j
end{equation}

for all real $a_0,a_1,a_2$.



So, the determinant of $M$ is $ge0$, which is equivalent to
begin{equation}
m_4ge m_4^*:=frac{m_2^3-2 m_1 m_3 m_2+m_0 m_3^2}{m_0 m_2-m_1^2}.
end{equation}

On the other hand, the difference between the left and right sides of your inequality (with $m_j$ in place of $chi(j)$) is increasing in $m_4$, and the value of this difference is $0$ at $m_4=m_4^*$. Thus, your inequality follows.






share|cite|improve this answer























  • Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
    – hopeless
    Nov 20 at 13:28













up vote
6
down vote



accepted







up vote
6
down vote



accepted






This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)ge R(x)$, where $L(x):=left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)$ and $R(x):=chi(3)chi(1)-chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz inequality, $R(x)>0$. Then for small enough $t>0$ we have $L(tx)=t^8L(x)<t^4R(x)=R(tx)$.





Added in response to the OP fixing the typo: The inequality is now true. Indeed, let $m_j:=chi(j)$. Without loss of generality, $m_0 m_2-m_1^2>0$, by the Cauchy--Schwarz inequality. The matrix $M:=[m_{i+j}]_{i,j=0}^2$ is positive semidefinite. This follows because
begin{equation}
0leint_0^1Big(sum_{i=0}^2 a_i, x(t)^iBig)^2f(t),dt=sum_{i,j=0}^2 m_{i+j}a_i a_j
end{equation}

for all real $a_0,a_1,a_2$.



So, the determinant of $M$ is $ge0$, which is equivalent to
begin{equation}
m_4ge m_4^*:=frac{m_2^3-2 m_1 m_3 m_2+m_0 m_3^2}{m_0 m_2-m_1^2}.
end{equation}

On the other hand, the difference between the left and right sides of your inequality (with $m_j$ in place of $chi(j)$) is increasing in $m_4$, and the value of this difference is $0$ at $m_4=m_4^*$. Thus, your inequality follows.






share|cite|improve this answer














This inequality is false in general, by homogeneity considerations. Indeed, it can be rewritten as $L(x)ge R(x)$, where $L(x):=left(chi(0)chi(2)-chi(1)^{2}right)left(chi(4)chi(2)-chi(3)^{2}right)$ and $R(x):=chi(3)chi(1)-chi(2)^{2}$. Take any non-constant positive $x$, so that, by the Cauchy--Schwarz inequality, $R(x)>0$. Then for small enough $t>0$ we have $L(tx)=t^8L(x)<t^4R(x)=R(tx)$.





Added in response to the OP fixing the typo: The inequality is now true. Indeed, let $m_j:=chi(j)$. Without loss of generality, $m_0 m_2-m_1^2>0$, by the Cauchy--Schwarz inequality. The matrix $M:=[m_{i+j}]_{i,j=0}^2$ is positive semidefinite. This follows because
begin{equation}
0leint_0^1Big(sum_{i=0}^2 a_i, x(t)^iBig)^2f(t),dt=sum_{i,j=0}^2 m_{i+j}a_i a_j
end{equation}

for all real $a_0,a_1,a_2$.



So, the determinant of $M$ is $ge0$, which is equivalent to
begin{equation}
m_4ge m_4^*:=frac{m_2^3-2 m_1 m_3 m_2+m_0 m_3^2}{m_0 m_2-m_1^2}.
end{equation}

On the other hand, the difference between the left and right sides of your inequality (with $m_j$ in place of $chi(j)$) is increasing in $m_4$, and the value of this difference is $0$ at $m_4=m_4^*$. Thus, your inequality follows.







share|cite|improve this answer














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edited Nov 20 at 15:12

























answered Nov 20 at 13:18









Iosif Pinelis

16.9k12157




16.9k12157












  • Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
    – hopeless
    Nov 20 at 13:28


















  • Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
    – hopeless
    Nov 20 at 13:28
















Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
– hopeless
Nov 20 at 13:28




Yes, sorry! There should have been a square on the second term. Sorry about this, I have edited the question. Thank you
– hopeless
Nov 20 at 13:28










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