Inequality in 3 variables (conjecture)
$begingroup$
Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$.
If $0<kleq 3+2sqrt{3}$, then $$frac{a}{b^2+k}+frac{b}{c^2+k}+frac{c}{a^2+k}geq frac{3}{1+k}$$
If $k=3+2sqrt{3}$, then equality occurs if $a=b=c=1$ or $a=0$, $b=1-sqrt{3}$ and $c=sqrt{3}$ or any cyclic permutation thereof. This inequality can be found in Vasile Cirtoaje's Discrete inequalities, Volume 4 - page 61. Any idea for this inequality? Thank you!
calculus algebra-precalculus inequality conjectures
asked Jul 14 '15 at 10:44
MarloMarlo
412
$endgroup$
add a comment |
$begingroup$
Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$.
If $0<kleq 3+2sqrt{3}$, then $$frac{a}{b^2+k}+frac{b}{c^2+k}+frac{c}{a^2+k}geq frac{3}{1+k}$$
If $k=3+2sqrt{3}$, then equality occurs if $a=b=c=1$ or $a=0$, $b=1-sqrt{3}$ and $c=sqrt{3}$ or any cyclic permutation thereof. This inequality can be found in Vasile Cirtoaje's Discrete inequalities, Volume 4 - page 61. Any idea for this inequality? Thank you!
calculus algebra-precalculus inequality conjectures
asked Jul 14 '15 at 10:44
MarloMarlo
412
$endgroup$
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
1
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20
add a comment |
$begingroup$
Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$.
If $0<kleq 3+2sqrt{3}$, then $$frac{a}{b^2+k}+frac{b}{c^2+k}+frac{c}{a^2+k}geq frac{3}{1+k}$$
If $k=3+2sqrt{3}$, then equality occurs if $a=b=c=1$ or $a=0$, $b=1-sqrt{3}$ and $c=sqrt{3}$ or any cyclic permutation thereof. This inequality can be found in Vasile Cirtoaje's Discrete inequalities, Volume 4 - page 61. Any idea for this inequality? Thank you!
calculus algebra-precalculus inequality conjectures
asked Jul 14 '15 at 10:44
MarloMarlo
412
$endgroup$
Let $a, b, c$ be nonnegative real numbers such that $a+b+c=3$.
If $0<kleq 3+2sqrt{3}$, then $$frac{a}{b^2+k}+frac{b}{c^2+k}+frac{c}{a^2+k}geq frac{3}{1+k}$$
If $k=3+2sqrt{3}$, then equality occurs if $a=b=c=1$ or $a=0$, $b=1-sqrt{3}$ and $c=sqrt{3}$ or any cyclic permutation thereof. This inequality can be found in Vasile Cirtoaje's Discrete inequalities, Volume 4 - page 61. Any idea for this inequality? Thank you!
calculus algebra-precalculus inequality conjectures
calculus algebra-precalculus inequality conjectures
asked Jul 14 '15 at 10:44
MarloMarlo
412
asked Jul 14 '15 at 10:44
MarloMarlo
412
asked Jul 14 '15 at 10:44
MarloMarlo
412
asked Jul 14 '15 at 10:44
MarloMarlo
412
asked Jul 14 '15 at 10:44
MarloMarlo
412
412
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
1
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20
add a comment |
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
1
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
1
1
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This has been solved. A proof was published in the "Gazeta matematica", a Romanian mathematics magazine for high school students:
Martin Bottesch. On a conjecture of Vasile Cirtoaje. Gazeta Matematica, Seria B, 12/2015.
http://ssmr.ro/gazeta/gmb/2015/12/articol.pdf
answered Jan 1 at 16:32
BBatturBBattur
111
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f1360575%2finequality-in-3-variables-conjecture%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This has been solved. A proof was published in the "Gazeta matematica", a Romanian mathematics magazine for high school students:
Martin Bottesch. On a conjecture of Vasile Cirtoaje. Gazeta Matematica, Seria B, 12/2015.
http://ssmr.ro/gazeta/gmb/2015/12/articol.pdf
answered Jan 1 at 16:32
BBatturBBattur
111
$endgroup$
add a comment |
$begingroup$
This has been solved. A proof was published in the "Gazeta matematica", a Romanian mathematics magazine for high school students:
Martin Bottesch. On a conjecture of Vasile Cirtoaje. Gazeta Matematica, Seria B, 12/2015.
http://ssmr.ro/gazeta/gmb/2015/12/articol.pdf
answered Jan 1 at 16:32
BBatturBBattur
111
$endgroup$
add a comment |
$begingroup$
This has been solved. A proof was published in the "Gazeta matematica", a Romanian mathematics magazine for high school students:
Martin Bottesch. On a conjecture of Vasile Cirtoaje. Gazeta Matematica, Seria B, 12/2015.
http://ssmr.ro/gazeta/gmb/2015/12/articol.pdf
answered Jan 1 at 16:32
BBatturBBattur
111
$endgroup$
This has been solved. A proof was published in the "Gazeta matematica", a Romanian mathematics magazine for high school students:
Martin Bottesch. On a conjecture of Vasile Cirtoaje. Gazeta Matematica, Seria B, 12/2015.
http://ssmr.ro/gazeta/gmb/2015/12/articol.pdf
answered Jan 1 at 16:32
BBatturBBattur
111
answered Jan 1 at 16:32
BBatturBBattur
111
answered Jan 1 at 16:32
BBatturBBattur
111
answered Jan 1 at 16:32
BBatturBBattur
111
111
add a comment |
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.
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%2f1360575%2finequality-in-3-variables-conjecture%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
$begingroup$
in this book is no proof?
$endgroup$
– Dr. Sonnhard Graubner
Jul 14 '15 at 10:44
1
$begingroup$
if this inequality is conjectured by vasile cirtoaje I doubt if someone could prove it in a short time.vasile cirtoaje is a master in inequalities and it would be very hard to tackle an inequality he did not proved.
$endgroup$
– user2838619
Jul 14 '15 at 11:13
$begingroup$
It is enough to prove only $k=3+2sqrt{3}$.
$endgroup$
– Takahiro Waki
Dec 22 '16 at 9:20