Under what conditions on $a,b,c$ is the sum $a^3 + b^3 + c^3$ strictly greater than $(a+b+c)^2$?












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I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.






elementary-number-theory







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  • $begingroup$
    One of the cases in which this holds is $a,b =0$ and $cgt1$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:24












  • $begingroup$
    The inequality holds $forall quad a,b,cgt 3$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:59


















0












$begingroup$


I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.






elementary-number-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    One of the cases in which this holds is $a,b =0$ and $cgt1$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:24












  • $begingroup$
    The inequality holds $forall quad a,b,cgt 3$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:59
















0












0








0





$begingroup$


I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.






elementary-number-theory







share|cite|improve this question









$endgroup$




I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.






elementary-number-theory




elementary-number-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Nov 28 '18 at 18:19









Matheus AndradeMatheus Andrade

1,168417




1,168417












  • $begingroup$
    One of the cases in which this holds is $a,b =0$ and $cgt1$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:24












  • $begingroup$
    The inequality holds $forall quad a,b,cgt 3$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:59




















  • $begingroup$
    One of the cases in which this holds is $a,b =0$ and $cgt1$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:24












  • $begingroup$
    The inequality holds $forall quad a,b,cgt 3$.
    $endgroup$
    – user612946
    Nov 28 '18 at 18:59


















$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24






$begingroup$
One of the cases in which this holds is $a,b =0$ and $cgt1$.
$endgroup$
– user612946
Nov 28 '18 at 18:24














$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59






$begingroup$
The inequality holds $forall quad a,b,cgt 3$.
$endgroup$
– user612946
Nov 28 '18 at 18:59












1 Answer
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$begingroup$

Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.



enter image description here






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    1 Answer
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    $begingroup$

    Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.



    enter image description here






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.



      enter image description here






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.



        enter image description here






        share|cite|improve this answer









        $endgroup$



        Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.



        enter image description here







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 28 '18 at 19:12









        Robert IsraelRobert Israel

        319k23209459




        319k23209459






























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