Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da} geq pi$












17












$begingroup$



If $a,b,c,d >0$, and $a+b+c+d=4$, prove that
$$a^{ab}+b^{bc}+c^{cd}+d^{da} geq pi.$$




I don't think Jensen's inequality will help here, but I think first determining where equality holds will be useful. Maybe taking the logarithm or exponential of both sides will also be useful, but I want to in the end get rid of the plus signs in order to simplify it.










share|cite|improve this question









$endgroup$












  • $begingroup$
    This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
    $endgroup$
    – Surb
    May 6 '16 at 20:54












  • $begingroup$
    May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
    $endgroup$
    – Wojowu
    May 6 '16 at 20:55












  • $begingroup$
    @Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
    $endgroup$
    – Puzzled417
    May 6 '16 at 20:59








  • 2




    $begingroup$
    The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
    $endgroup$
    – Patrick Stevens
    May 6 '16 at 21:26






  • 2




    $begingroup$
    An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
    $endgroup$
    – Yuriy S
    Apr 25 '18 at 10:04
















17












$begingroup$



If $a,b,c,d >0$, and $a+b+c+d=4$, prove that
$$a^{ab}+b^{bc}+c^{cd}+d^{da} geq pi.$$




I don't think Jensen's inequality will help here, but I think first determining where equality holds will be useful. Maybe taking the logarithm or exponential of both sides will also be useful, but I want to in the end get rid of the plus signs in order to simplify it.










share|cite|improve this question









$endgroup$












  • $begingroup$
    This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
    $endgroup$
    – Surb
    May 6 '16 at 20:54












  • $begingroup$
    May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
    $endgroup$
    – Wojowu
    May 6 '16 at 20:55












  • $begingroup$
    @Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
    $endgroup$
    – Puzzled417
    May 6 '16 at 20:59








  • 2




    $begingroup$
    The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
    $endgroup$
    – Patrick Stevens
    May 6 '16 at 21:26






  • 2




    $begingroup$
    An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
    $endgroup$
    – Yuriy S
    Apr 25 '18 at 10:04














17












17








17


14



$begingroup$



If $a,b,c,d >0$, and $a+b+c+d=4$, prove that
$$a^{ab}+b^{bc}+c^{cd}+d^{da} geq pi.$$




I don't think Jensen's inequality will help here, but I think first determining where equality holds will be useful. Maybe taking the logarithm or exponential of both sides will also be useful, but I want to in the end get rid of the plus signs in order to simplify it.










share|cite|improve this question









$endgroup$





If $a,b,c,d >0$, and $a+b+c+d=4$, prove that
$$a^{ab}+b^{bc}+c^{cd}+d^{da} geq pi.$$




I don't think Jensen's inequality will help here, but I think first determining where equality holds will be useful. Maybe taking the logarithm or exponential of both sides will also be useful, but I want to in the end get rid of the plus signs in order to simplify it.







inequality






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked May 6 '16 at 20:49









Puzzled417Puzzled417

3,105527




3,105527












  • $begingroup$
    This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
    $endgroup$
    – Surb
    May 6 '16 at 20:54












  • $begingroup$
    May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
    $endgroup$
    – Wojowu
    May 6 '16 at 20:55












  • $begingroup$
    @Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
    $endgroup$
    – Puzzled417
    May 6 '16 at 20:59








  • 2




    $begingroup$
    The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
    $endgroup$
    – Patrick Stevens
    May 6 '16 at 21:26






  • 2




    $begingroup$
    An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
    $endgroup$
    – Yuriy S
    Apr 25 '18 at 10:04


















  • $begingroup$
    This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
    $endgroup$
    – Surb
    May 6 '16 at 20:54












  • $begingroup$
    May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
    $endgroup$
    – Wojowu
    May 6 '16 at 20:55












  • $begingroup$
    @Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
    $endgroup$
    – Puzzled417
    May 6 '16 at 20:59








  • 2




    $begingroup$
    The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
    $endgroup$
    – Patrick Stevens
    May 6 '16 at 21:26






  • 2




    $begingroup$
    An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
    $endgroup$
    – Yuriy S
    Apr 25 '18 at 10:04
















$begingroup$
This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
$endgroup$
– Surb
May 6 '16 at 20:54






$begingroup$
This reminds me the inequality $a^a+b^b>a^b+b^a$ from Wikipédia. Not sure if it help though...
$endgroup$
– Surb
May 6 '16 at 20:54














$begingroup$
May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
$endgroup$
– Wojowu
May 6 '16 at 20:55






$begingroup$
May I ask how you came across this problem? Also, do you know whether the constant $pi$ is optimal?
$endgroup$
– Wojowu
May 6 '16 at 20:55














$begingroup$
@Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
$endgroup$
– Puzzled417
May 6 '16 at 20:59






$begingroup$
@Wojowu It is from a problem solving website. Yes, I believe equality is achieved.
$endgroup$
– Puzzled417
May 6 '16 at 20:59






2




2




$begingroup$
The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
$endgroup$
– Patrick Stevens
May 6 '16 at 21:26




$begingroup$
The best I've got out of Mathematica is 3.1605859174508652189…, using NMinimize[{(a^2)^(a^2 b^2) + (b^2)^(b^2 c^2) + (c^2)^(c^2 d^2) + (d^2)^(d^2 a^2), a != 0 && b != 0 && c != 0 && d != 0 && a^2 + b^2 + c^2 + d^2 == 4}, {a, b, c, d}, WorkingPrecision -> 100]
$endgroup$
– Patrick Stevens
May 6 '16 at 21:26




2




2




$begingroup$
An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
$endgroup$
– Yuriy S
Apr 25 '18 at 10:04




$begingroup$
An interesting inequality, but even without the numerical evidence for the bound being around $3.16$ there doesn't seem to be any reason for $pi$ to feature here. Seems arbitrary
$endgroup$
– Yuriy S
Apr 25 '18 at 10:04










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