Bound sum by integral
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I would like to prove the result
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq8int_{s/4}^tH^{1/2}(x)dx,
$$
where $H:(0,t]tomathbb{R}^+$ is a decreasing function and $0<s<t$ with $delta_0=t$, $delta_{j+1}=sveesup{xleqdelta_j/2:H(x)geq4H(delta_j)}$ for $jgeq0$ and $N=min{j:delta_j=s}$.
I try to prove the result by showing
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq constsum_{j=0}^N(delta_{j+1}-delta_{j+2})H^{1/2}(delta_{j+1})
$$
so that the right hand side can be bounded by the desired integral. But I had some issues in obtainting this inequality. I was wondering whether my idea is correct. Thanks in advance.
real-analysis calculus integration
asked Dec 2 at 20:25
M.Shen
263
add a comment |
up vote
0
down vote
favorite
I would like to prove the result
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq8int_{s/4}^tH^{1/2}(x)dx,
$$
where $H:(0,t]tomathbb{R}^+$ is a decreasing function and $0<s<t$ with $delta_0=t$, $delta_{j+1}=sveesup{xleqdelta_j/2:H(x)geq4H(delta_j)}$ for $jgeq0$ and $N=min{j:delta_j=s}$.
I try to prove the result by showing
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq constsum_{j=0}^N(delta_{j+1}-delta_{j+2})H^{1/2}(delta_{j+1})
$$
so that the right hand side can be bounded by the desired integral. But I had some issues in obtainting this inequality. I was wondering whether my idea is correct. Thanks in advance.
real-analysis calculus integration
asked Dec 2 at 20:25
M.Shen
263
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I would like to prove the result
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq8int_{s/4}^tH^{1/2}(x)dx,
$$
where $H:(0,t]tomathbb{R}^+$ is a decreasing function and $0<s<t$ with $delta_0=t$, $delta_{j+1}=sveesup{xleqdelta_j/2:H(x)geq4H(delta_j)}$ for $jgeq0$ and $N=min{j:delta_j=s}$.
I try to prove the result by showing
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq constsum_{j=0}^N(delta_{j+1}-delta_{j+2})H^{1/2}(delta_{j+1})
$$
so that the right hand side can be bounded by the desired integral. But I had some issues in obtainting this inequality. I was wondering whether my idea is correct. Thanks in advance.
real-analysis calculus integration
asked Dec 2 at 20:25
M.Shen
263
I would like to prove the result
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq8int_{s/4}^tH^{1/2}(x)dx,
$$
where $H:(0,t]tomathbb{R}^+$ is a decreasing function and $0<s<t$ with $delta_0=t$, $delta_{j+1}=sveesup{xleqdelta_j/2:H(x)geq4H(delta_j)}$ for $jgeq0$ and $N=min{j:delta_j=s}$.
I try to prove the result by showing
$$
sum_{j=0}^Ndelta_jH^{1/2}(delta_{j+1})leq constsum_{j=0}^N(delta_{j+1}-delta_{j+2})H^{1/2}(delta_{j+1})
$$
so that the right hand side can be bounded by the desired integral. But I had some issues in obtainting this inequality. I was wondering whether my idea is correct. Thanks in advance.
real-analysis calculus integration
real-analysis calculus integration
asked Dec 2 at 20:25
M.Shen
263
asked Dec 2 at 20:25
M.Shen
263
asked Dec 2 at 20:25
M.Shen
263
asked Dec 2 at 20:25
M.Shen
263
asked Dec 2 at 20:25
M.Shen
263
263
add a comment |
add a comment |
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