Obtaining an upper bound for the integral
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For $x,y in mathbb{R}^n$ and for a fixed $p in mathbb{R}$, I have the integral
begin{equation}
I(x)=int_{mathbb{R}^n} frac{langle x+ty rangle ^p}{langle y rangle ^q}dy
end{equation}
where $langle x rangle =(1+|x|^2)^{1/2}$ and $0<t<1$, also that $q in mathbb{R}$ can be taken as large as possible to guarantee the convergence.
Now, is it possible to show that there is $C>0$
begin{equation}
I(x) leq Clangle x rangle ^p?
end{equation}
I am able to show the above inequality when $p geq 0$. I need help in showing the inequality for $p < 0$. Thank you in advance.
real-analysis integration
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
add a comment |
up vote
0
down vote
favorite
For $x,y in mathbb{R}^n$ and for a fixed $p in mathbb{R}$, I have the integral
begin{equation}
I(x)=int_{mathbb{R}^n} frac{langle x+ty rangle ^p}{langle y rangle ^q}dy
end{equation}
where $langle x rangle =(1+|x|^2)^{1/2}$ and $0<t<1$, also that $q in mathbb{R}$ can be taken as large as possible to guarantee the convergence.
Now, is it possible to show that there is $C>0$
begin{equation}
I(x) leq Clangle x rangle ^p?
end{equation}
I am able to show the above inequality when $p geq 0$. I need help in showing the inequality for $p < 0$. Thank you in advance.
real-analysis integration
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For $x,y in mathbb{R}^n$ and for a fixed $p in mathbb{R}$, I have the integral
begin{equation}
I(x)=int_{mathbb{R}^n} frac{langle x+ty rangle ^p}{langle y rangle ^q}dy
end{equation}
where $langle x rangle =(1+|x|^2)^{1/2}$ and $0<t<1$, also that $q in mathbb{R}$ can be taken as large as possible to guarantee the convergence.
Now, is it possible to show that there is $C>0$
begin{equation}
I(x) leq Clangle x rangle ^p?
end{equation}
I am able to show the above inequality when $p geq 0$. I need help in showing the inequality for $p < 0$. Thank you in advance.
real-analysis integration
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
For $x,y in mathbb{R}^n$ and for a fixed $p in mathbb{R}$, I have the integral
begin{equation}
I(x)=int_{mathbb{R}^n} frac{langle x+ty rangle ^p}{langle y rangle ^q}dy
end{equation}
where $langle x rangle =(1+|x|^2)^{1/2}$ and $0<t<1$, also that $q in mathbb{R}$ can be taken as large as possible to guarantee the convergence.
Now, is it possible to show that there is $C>0$
begin{equation}
I(x) leq Clangle x rangle ^p?
end{equation}
I am able to show the above inequality when $p geq 0$. I need help in showing the inequality for $p < 0$. Thank you in advance.
real-analysis integration
real-analysis integration
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
edited Nov 17 at 13:02
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
asked Nov 17 at 10:33
Rahul Raju Pattar
354110
354110
add a comment |
add a comment |
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