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If $a$, $b$ and $c$ are positive real numbers, find the minimum value of $\sqrt { \frac { a }{ b+c } } +\sqrt [ 3 ]{ \frac { b }{ c+a } } +\sqrt [ 4 ]{ \frac { c }{ a+b } } $.

I am not able to progress in this problem.I tried applying AM-GM,Cauchy,Weighted AM-GM,etc. but none seem to provide fruitful results. Please help.

Source: A collection of problems which couldn't be solved by any teacher of my school.

Thanks.

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  • $\begingroup$ Always greater than 1 ,what we put $a=b=c$ it will be minimum $\endgroup$ – Archis Welankar Dec 26 '15 at 12:21
  • $\begingroup$ Letting $c\to0$ and $a=b$ we have $S=2,$ which is lesser than $S=2.34^+$ for when $a=b=c.$ $\endgroup$ – Lucian Dec 26 '15 at 13:27
  • $\begingroup$ i got this as the searched Minimum $1.8898816003888315111930771751475078051911930043602,$ $\endgroup$ – Dr. Sonnhard Graubner Dec 26 '15 at 13:50
  • $\begingroup$ Dear Dr. Sonnhard Graubner! The minimum does not exist. An infimum is $\frac{3}{\sqrt[3]4}$. $\endgroup$ – Michael Rozenberg Dec 27 '15 at 13:43

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