Given $n_1 ge d_1 ge 0$, $n_2 ge d_2 ge 0$ and $d_1 + d_2 gt 0$, how do I show that…
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Given $n_1 ge d_1 ge 0$, $n_2 ge d_2 ge 0$ and $d_1 + d_2 gt 0$, how do I show that
$$Big( frac{n_1}{d_1} Big)^{d_1} Big( frac{n_2 }{d_2} Big)^{d_2} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$$
I have managed to split this into 3 cases:
Case 1: $n_1d_2 = d_1n_2$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which satisfies the expression above.
My problem here is with cases 2 and 3.
For example,
Case 2: $n_1d_2 le n_2d_1$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} ge Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which's where I am conflicted.
How can I show that this conflict works itself out to satisfy th main expression?
Thank you.
combinatorics algebra-precalculus discrete-mathematics
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Given $n_1 ge d_1 ge 0$, $n_2 ge d_2 ge 0$ and $d_1 + d_2 gt 0$, how do I show that
$$Big( frac{n_1}{d_1} Big)^{d_1} Big( frac{n_2 }{d_2} Big)^{d_2} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$$
I have managed to split this into 3 cases:
Case 1: $n_1d_2 = d_1n_2$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which satisfies the expression above.
My problem here is with cases 2 and 3.
For example,
Case 2: $n_1d_2 le n_2d_1$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} ge Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which's where I am conflicted.
How can I show that this conflict works itself out to satisfy th main expression?
Thank you.
combinatorics algebra-precalculus discrete-mathematics
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given $n_1 ge d_1 ge 0$, $n_2 ge d_2 ge 0$ and $d_1 + d_2 gt 0$, how do I show that
$$Big( frac{n_1}{d_1} Big)^{d_1} Big( frac{n_2 }{d_2} Big)^{d_2} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$$
I have managed to split this into 3 cases:
Case 1: $n_1d_2 = d_1n_2$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which satisfies the expression above.
My problem here is with cases 2 and 3.
For example,
Case 2: $n_1d_2 le n_2d_1$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} ge Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which's where I am conflicted.
How can I show that this conflict works itself out to satisfy th main expression?
Thank you.
combinatorics algebra-precalculus discrete-mathematics
Given $n_1 ge d_1 ge 0$, $n_2 ge d_2 ge 0$ and $d_1 + d_2 gt 0$, how do I show that
$$Big( frac{n_1}{d_1} Big)^{d_1} Big( frac{n_2 }{d_2} Big)^{d_2} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$$
I have managed to split this into 3 cases:
Case 1: $n_1d_2 = d_1n_2$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} = Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which satisfies the expression above.
My problem here is with cases 2 and 3.
For example,
Case 2: $n_1d_2 le n_2d_1$
This leads to $Big( frac{n_1}{d_1} Big)^{d_1} le Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_1}$, and also leads to $Big( frac{n_2}{d_2} Big)^{d_2} ge Big(frac{(n_1 + n_2)}{d_1 + d_2}Big)^{d_2}$, which's where I am conflicted.
How can I show that this conflict works itself out to satisfy th main expression?
Thank you.
combinatorics algebra-precalculus discrete-mathematics
combinatorics algebra-precalculus discrete-mathematics
edited Nov 18 at 12:39
asked Nov 18 at 9:43
user617040
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You can show your inequality by taking the logarithm and using the concavity of $log$:
$log(lambda x + (1-lambda)y) geq lambda log x + (1-lambda) log y$ for $lambda in [0,1]$
Set
- $lambda = frac{d_1}{d_1+d_2} Rightarrow 1-lambda = frac{d_2}{d_1+d_2}$
- $x= frac{n_1}{d_1}$
- $y= frac{n_2}{d_2}$
Taking the logarithm on the right side of your inequality gives
$$d_1 log frac{n_1}{d_1} + d_2 log frac{n_2}{d_2} le (d_1+d_2) log left( frac{n_1}{d_1+d_2} + frac{n_2}{d_1+d_2}right)$$
Can you take it from here?
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
You can show your inequality by taking the logarithm and using the concavity of $log$:
$log(lambda x + (1-lambda)y) geq lambda log x + (1-lambda) log y$ for $lambda in [0,1]$
Set
- $lambda = frac{d_1}{d_1+d_2} Rightarrow 1-lambda = frac{d_2}{d_1+d_2}$
- $x= frac{n_1}{d_1}$
- $y= frac{n_2}{d_2}$
Taking the logarithm on the right side of your inequality gives
$$d_1 log frac{n_1}{d_1} + d_2 log frac{n_2}{d_2} le (d_1+d_2) log left( frac{n_1}{d_1+d_2} + frac{n_2}{d_1+d_2}right)$$
Can you take it from here?
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
add a comment |
up vote
0
down vote
accepted
You can show your inequality by taking the logarithm and using the concavity of $log$:
$log(lambda x + (1-lambda)y) geq lambda log x + (1-lambda) log y$ for $lambda in [0,1]$
Set
- $lambda = frac{d_1}{d_1+d_2} Rightarrow 1-lambda = frac{d_2}{d_1+d_2}$
- $x= frac{n_1}{d_1}$
- $y= frac{n_2}{d_2}$
Taking the logarithm on the right side of your inequality gives
$$d_1 log frac{n_1}{d_1} + d_2 log frac{n_2}{d_2} le (d_1+d_2) log left( frac{n_1}{d_1+d_2} + frac{n_2}{d_1+d_2}right)$$
Can you take it from here?
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
You can show your inequality by taking the logarithm and using the concavity of $log$:
$log(lambda x + (1-lambda)y) geq lambda log x + (1-lambda) log y$ for $lambda in [0,1]$
Set
- $lambda = frac{d_1}{d_1+d_2} Rightarrow 1-lambda = frac{d_2}{d_1+d_2}$
- $x= frac{n_1}{d_1}$
- $y= frac{n_2}{d_2}$
Taking the logarithm on the right side of your inequality gives
$$d_1 log frac{n_1}{d_1} + d_2 log frac{n_2}{d_2} le (d_1+d_2) log left( frac{n_1}{d_1+d_2} + frac{n_2}{d_1+d_2}right)$$
Can you take it from here?
You can show your inequality by taking the logarithm and using the concavity of $log$:
$log(lambda x + (1-lambda)y) geq lambda log x + (1-lambda) log y$ for $lambda in [0,1]$
Set
- $lambda = frac{d_1}{d_1+d_2} Rightarrow 1-lambda = frac{d_2}{d_1+d_2}$
- $x= frac{n_1}{d_1}$
- $y= frac{n_2}{d_2}$
Taking the logarithm on the right side of your inequality gives
$$d_1 log frac{n_1}{d_1} + d_2 log frac{n_2}{d_2} le (d_1+d_2) log left( frac{n_1}{d_1+d_2} + frac{n_2}{d_1+d_2}right)$$
Can you take it from here?
edited Nov 18 at 10:24
answered Nov 18 at 10:16
trancelocation
8,4571520
8,4571520
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
add a comment |
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
I can, yes! Thank you for your response! Although I'm wondering if there's any simpler way I could try to reach the same conclusion.
– user617040
Nov 18 at 10:53
add a comment |
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