product of sines is less than or equal to sine of arithmetic mean
$begingroup$
Let $x=frac{sum_{i=1}^nx_i}{n}$ and let the angles $x_iin(0,pi)$. Then, is the following inequality true?
$$prod_{i=1}^nsin(x_i)lesin^nx$$
I think yes, by analogy with the AM-GM inequality. But, the application is not direct. Does convexity of the sine function in the interval have a role here? Does Jensen inequality work here? Thanks beforehand.
algebra-precalculus trigonometry inequality
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
$endgroup$
add a comment |
$begingroup$
Let $x=frac{sum_{i=1}^nx_i}{n}$ and let the angles $x_iin(0,pi)$. Then, is the following inequality true?
$$prod_{i=1}^nsin(x_i)lesin^nx$$
I think yes, by analogy with the AM-GM inequality. But, the application is not direct. Does convexity of the sine function in the interval have a role here? Does Jensen inequality work here? Thanks beforehand.
algebra-precalculus trigonometry inequality
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
$endgroup$
1
$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24
add a comment |
$begingroup$
Let $x=frac{sum_{i=1}^nx_i}{n}$ and let the angles $x_iin(0,pi)$. Then, is the following inequality true?
$$prod_{i=1}^nsin(x_i)lesin^nx$$
I think yes, by analogy with the AM-GM inequality. But, the application is not direct. Does convexity of the sine function in the interval have a role here? Does Jensen inequality work here? Thanks beforehand.
algebra-precalculus trigonometry inequality
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
$endgroup$
Let $x=frac{sum_{i=1}^nx_i}{n}$ and let the angles $x_iin(0,pi)$. Then, is the following inequality true?
$$prod_{i=1}^nsin(x_i)lesin^nx$$
I think yes, by analogy with the AM-GM inequality. But, the application is not direct. Does convexity of the sine function in the interval have a role here? Does Jensen inequality work here? Thanks beforehand.
algebra-precalculus trigonometry inequality
algebra-precalculus trigonometry inequality
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
edited Jan 2 at 15:26
vidyarthi
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
asked Jan 2 at 15:16
vidyarthividyarthi
3,0731833
3,0731833
1
$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24
add a comment |
1
$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24
1
1
$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24
$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24
add a comment |
1 Answer
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$begingroup$
The sine function is concave on $(0, pi)$, therefore
$$
sin x ge frac 1n sum_{i=1}^n sin x_i , .
$$
Using the AM-GM inequality, this expression is
$$
ge left (prod_{i=1}^nsin(x_i)right) ^{1/n} , .
$$
You can also apply Jensen's inequality to the (concave) function $log circ sin$ to obtain the same result.
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
$endgroup$
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The sine function is concave on $(0, pi)$, therefore
$$
sin x ge frac 1n sum_{i=1}^n sin x_i , .
$$
Using the AM-GM inequality, this expression is
$$
ge left (prod_{i=1}^nsin(x_i)right) ^{1/n} , .
$$
You can also apply Jensen's inequality to the (concave) function $log circ sin$ to obtain the same result.
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
$endgroup$
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
add a comment |
$begingroup$
The sine function is concave on $(0, pi)$, therefore
$$
sin x ge frac 1n sum_{i=1}^n sin x_i , .
$$
Using the AM-GM inequality, this expression is
$$
ge left (prod_{i=1}^nsin(x_i)right) ^{1/n} , .
$$
You can also apply Jensen's inequality to the (concave) function $log circ sin$ to obtain the same result.
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
$endgroup$
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
add a comment |
$begingroup$
The sine function is concave on $(0, pi)$, therefore
$$
sin x ge frac 1n sum_{i=1}^n sin x_i , .
$$
Using the AM-GM inequality, this expression is
$$
ge left (prod_{i=1}^nsin(x_i)right) ^{1/n} , .
$$
You can also apply Jensen's inequality to the (concave) function $log circ sin$ to obtain the same result.
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
$endgroup$
The sine function is concave on $(0, pi)$, therefore
$$
sin x ge frac 1n sum_{i=1}^n sin x_i , .
$$
Using the AM-GM inequality, this expression is
$$
ge left (prod_{i=1}^nsin(x_i)right) ^{1/n} , .
$$
You can also apply Jensen's inequality to the (concave) function $log circ sin$ to obtain the same result.
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
edited Jan 2 at 15:47
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
answered Jan 2 at 15:26
Martin RMartin R
30.5k33558
30.5k33558
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
add a comment |
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
$begingroup$
I actually edited my question. But your answer applies to my previous mistaken question as well!
$endgroup$
– vidyarthi
Jan 2 at 15:29
add a comment |
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$begingroup$
Hint: apply logarithm on both side. Btw sine is concave on $(0, pin)$, not convex.
$endgroup$
– user251257
Jan 2 at 15:24