March 2023
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In the Math Corner of my March 2023 Diary I offered the following brainteaser.
Let a, b, and c be positive real numbers such that a² + b² + c² ≤ 3. Prove that
√(1 + a) + √(1 + b) + √(1 + c) ≥ (√2)(a + b + c)² / 3
When does equality occur?
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• Solution
Difficult! Obviously equality occurs when a = b = c = 1; but the general proof eluded me.
Two readers with better math than mine did prove the inequality, but they used advanced techniques.
Reader A used Lagrange multipliers, a neat trick in calculus. (However, several users of Twitter, where Reader A posted his proof, found a flaw in it.)
Reader B plunged deeper, resorting to Schur minimization.
At mid-July the sense of the meeting is that a tweeter identifying only as "Hedi H," Twitter handle "@HHHHdi," has come up with the nearest yet to a solution by elementary algebra, although "elementary" here translates as "undergraduate math level."
Hedi H's result depends on two standard inequalities, those of Jensen and Cauchy-Schwarz. Both are highly serviceable in any problem involving algebraic inequalities; the trick is to find how to apply them in some particular case.
I'm curious to see what approach will be taken by the person who posed the problem, when Math Horizons publishes the proof later this year (?).