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I encountered the following problem in my textbook-

Let $a, b, c$ be three arbitrary real numbers. Denote $$ x = \sqrt{b^2-bc+c^2}, y = \sqrt{c^2-ca+a^2}, z = \sqrt{a^2-ab+b^2} $$ Prove that $$ xy+yz+zx \ge a^2+b^2+c^2 $$

Textbook's Solution:
Rewrite x,y in the following forms $$ x = \sqrt{{3c^2\over 4}+\left(b-{c\over 2}\right)^2}, y = \sqrt{{3c^2\over 4}+\left(a-{c\over 2}\right)^2} $$
According to Cauchy-Schwarz inequality, we conclude $$ xy \ge {3c^2\over 4}+{1\over 4}\left( 2b-c\right)\left(2a-c \right) $$
which implies
$$ \sum_{cyc}xy \ge {3\over 4}\sum_{cyc}c^2 +{1\over 4}\sum_{cyc}\left( 2b-c\right)\left(2a-c \right) = \sum_{cyc}a^2. $$ My Approach:
From the given values of $x$ and $y$- $$ xy = \sqrt{\left( c^2-bc+b^2\right) \left(c^2-ca+a^2 \right)} \ge c^2+c\sqrt{ab}+ab $$ So, $$xy+yz+zx \ge \sum_{cyc}a^2+\sum_{cyc}a\sqrt{bc}+\sum_{cyc}bc$$ And it rests to prove that- $$ \sum_{cyc}a^2+\sum_{cyc}a\sqrt{bc}+\sum_{cyc}bc \ge \sum_{cyc}a^2 $$ $$ \sum_{cyc}a\sqrt{bc}+\sum_{cyc}bc \ge 0 $$ Now my question is that how to prove this? Or are we done?
And also if both approachs are correct, what is the equality case?
The equality case is not given in the textbook so I ask.
Thanks!

VIVID
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Book Of Flames
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  • You can obtain equality cases from where you are applying C-S (in textbook's solution). – River Li Oct 09 '20 at 09:58
  • @RiverLi Thanks, I got it a few days ago in some other post of mine. But what when the equality case needs to be noticed before attempting the solution? – Book Of Flames Oct 09 '20 at 10:11
  • In my opinion, equality cases are important to design the coefficients when we are applying C-S or Holder or using bounds, etc. For example, in my previous answer, we design the bound $\sqrt{x} \ge f(x)$ according to the equality cases $x = 0, x=1, x=\frac{36}{25}$. https://math.stackexchange.com/questions/2057006/prove-that-sum-limits-cyc-sqrt-frac8ab8ac9bc2bcb2c-geq5/3446826#3446826 – River Li Oct 09 '20 at 11:55

1 Answers1

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Your first step is wrong.

Try $a=b=c$.

Id est, you got a right inequality, but after a wrong step and the proof of your last inequality is not relevant already.

Michael Rozenberg
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