Inequality in 3 variables (conjecture)












2












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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







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$endgroup$












  • $begingroup$
    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
















2












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    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














2












2








2


2



$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











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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


















  • $begingroup$
    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










1 Answer
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$begingroup$

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






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    1 Answer
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    oldest

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    active

    oldest

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    active

    oldest

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    $begingroup$

    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






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      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






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        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






        share|cite|improve this answer









        $endgroup$



        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







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 1 at 16:32









        BBatturBBattur

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