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For example- $$ \frac{a}{c+a-b} + \frac{b}{a+b-c} + \frac{c}{b+c-a} \geq 3. $$ This can be proven if take a=b=c Is there any theorem regarding this?

Franklin Pezzuti Dyer
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  • Think about "${a\over b}\le 2$" ... – Noah Schweber Jun 26 '18 at 19:09
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    $(a-b)^2\le 3$ is true for $a=b$, but that does not prove the (false) statement that it is true for all $a,b$ – Hagen von Eitzen Jun 26 '18 at 19:09
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    It is true that in many classical and competition-type inequalities equality holds when all variables are equal. It is helpful to know when equality holds- it can be used as a hint for how to proceed with the proof. – Michal Adamaszek Jun 26 '18 at 19:10
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    This might be part of a proof. You may be able to show, for example, that a function (typically one that’s invariant under a permutation of its variables) attains a global minimum (or maximum) for a given sum of the variables when the variables are equal. Here’s a somewhat-related example: https://math.stackexchange.com/questions/1695727/inequality-involving-four-numbers/1696223#1696223 – Steve Kass Jun 26 '18 at 19:20
  • Are the $a,b,c$ assumed to be sides of a triangle? – Dr. Sonnhard Graubner Jun 28 '19 at 14:36
  • @Dr.SonnhardGraubner No they are assumed to be positive reals. –  Jun 29 '19 at 16:05

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