$-loglvert P(x)rvert$ is convex on an interval between two consecutive roots
up vote
0
down vote
favorite
Let $P(x)=a_nx^n+dots+a_0$ ($n>1,a_nneq0$) be a polynomial with real coefficients that has only real roots. Then the function $f(x)=-loglvert P(x)rvert$ is convex on any open interval between two consecutive roots of $P(x)$.
I have tried to use $f''geq 0$ and induction on $n$ without much success. Thanks for any help!
real-analysis
edited Nov 18 at 19:57
Bernard
117k637109
asked Nov 18 at 19:25
user64066
1,626715
add a comment |
up vote
0
down vote
favorite
Let $P(x)=a_nx^n+dots+a_0$ ($n>1,a_nneq0$) be a polynomial with real coefficients that has only real roots. Then the function $f(x)=-loglvert P(x)rvert$ is convex on any open interval between two consecutive roots of $P(x)$.
I have tried to use $f''geq 0$ and induction on $n$ without much success. Thanks for any help!
real-analysis
edited Nov 18 at 19:57
Bernard
117k637109
asked Nov 18 at 19:25
user64066
1,626715
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $P(x)=a_nx^n+dots+a_0$ ($n>1,a_nneq0$) be a polynomial with real coefficients that has only real roots. Then the function $f(x)=-loglvert P(x)rvert$ is convex on any open interval between two consecutive roots of $P(x)$.
I have tried to use $f''geq 0$ and induction on $n$ without much success. Thanks for any help!
real-analysis
edited Nov 18 at 19:57
Bernard
117k637109
asked Nov 18 at 19:25
user64066
1,626715
Let $P(x)=a_nx^n+dots+a_0$ ($n>1,a_nneq0$) be a polynomial with real coefficients that has only real roots. Then the function $f(x)=-loglvert P(x)rvert$ is convex on any open interval between two consecutive roots of $P(x)$.
I have tried to use $f''geq 0$ and induction on $n$ without much success. Thanks for any help!
real-analysis
real-analysis
edited Nov 18 at 19:57
Bernard
117k637109
asked Nov 18 at 19:25
user64066
1,626715
edited Nov 18 at 19:57
Bernard
117k637109
asked Nov 18 at 19:25
user64066
1,626715
edited Nov 18 at 19:57
Bernard
117k637109
edited Nov 18 at 19:57
Bernard
117k637109
edited Nov 18 at 19:57
Bernard
117k637109
117k637109
asked Nov 18 at 19:25
user64066
1,626715
asked Nov 18 at 19:25
user64066
1,626715
asked Nov 18 at 19:25
user64066
1,626715
1,626715
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
By factoring out the roots, we may write $P(x)=prod_i (x-r_i)$, where we have used the assumption that all roots are real. Then $f(x)=-sum_ilog(|x-r_i|)$. Since the sum of convex functions is convex, it only remains to check that $-log(|x-r_i|)$ is convex on any interval not containing one of the roots, which can be done by verifying that the second derivative is positive.
answered Nov 18 at 22:09
Mike Hawk
1,402110
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
By factoring out the roots, we may write $P(x)=prod_i (x-r_i)$, where we have used the assumption that all roots are real. Then $f(x)=-sum_ilog(|x-r_i|)$. Since the sum of convex functions is convex, it only remains to check that $-log(|x-r_i|)$ is convex on any interval not containing one of the roots, which can be done by verifying that the second derivative is positive.
answered Nov 18 at 22:09
Mike Hawk
1,402110
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
add a comment |
up vote
1
down vote
accepted
By factoring out the roots, we may write $P(x)=prod_i (x-r_i)$, where we have used the assumption that all roots are real. Then $f(x)=-sum_ilog(|x-r_i|)$. Since the sum of convex functions is convex, it only remains to check that $-log(|x-r_i|)$ is convex on any interval not containing one of the roots, which can be done by verifying that the second derivative is positive.
answered Nov 18 at 22:09
Mike Hawk
1,402110
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
By factoring out the roots, we may write $P(x)=prod_i (x-r_i)$, where we have used the assumption that all roots are real. Then $f(x)=-sum_ilog(|x-r_i|)$. Since the sum of convex functions is convex, it only remains to check that $-log(|x-r_i|)$ is convex on any interval not containing one of the roots, which can be done by verifying that the second derivative is positive.
answered Nov 18 at 22:09
Mike Hawk
1,402110
By factoring out the roots, we may write $P(x)=prod_i (x-r_i)$, where we have used the assumption that all roots are real. Then $f(x)=-sum_ilog(|x-r_i|)$. Since the sum of convex functions is convex, it only remains to check that $-log(|x-r_i|)$ is convex on any interval not containing one of the roots, which can be done by verifying that the second derivative is positive.
answered Nov 18 at 22:09
Mike Hawk
1,402110
answered Nov 18 at 22:09
Mike Hawk
1,402110
answered Nov 18 at 22:09
Mike Hawk
1,402110
answered Nov 18 at 22:09
Mike Hawk
1,402110
1,402110
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
add a comment |
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
$P(x)=a_nprod_i (x-r_i)$, but that does not make a difference
– Paul Frost
Nov 18 at 22:47
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004006%2flog-lvert-px-rvert-is-convex-on-an-interval-between-two-consecutive-roots%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
