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Abi-Khuzam Inequality


The inequality

 sinAsinBsinC<=((3sqrt(3))/(2pi))^3ABC,

where A, B, and C are the vertex angles of a triangle. The maximum is reached for an equilateral triangle (and therefore at A=B=C=pi/3) and has numerical value 0.56559562463... (OEIS A127205).

The inequality was proven by Abi-Khuzam (1974), also considered by Klamkin (1977), and mentioned by Flanders (1978) as "an interesting related result" for the product sinAsinBsinC of the three angles A, B, and C of a triangle.


See also

Brocard Angle

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References

Abi-Khuzam, F. "Proof of Yff's Conjecture on the Brocard Angle of a Triangle." Elem. Math. 29, 141-142, 1974.Abi-Khuzam, F. F. and Boghossian, A. B. "Some Recent Geometric Inequalities." Amer. Math. Monthly 96, 576-589, 1989.Flanders, H. "Review of 'Problems and Theorems in Analysis,' by Pólya and Szegö." Bull. Amer. Math. Soc. 86, 53-62, 1978.Klamkin, M. S. "On Yff's Inequality for the Brocard Angle of a Triangle." Elem. Math. 32, 188, 1977.Kuipers, L. "Extension of Abi-Khuzam's Inequality to More than Four Angles." Nieuw Tijdschr. Wisk. 69, 166-169, 1981-1982.Sloane, N. J. A. Sequence A127205 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Abi-Khuzam Inequality

Cite this as:

Weisstein, Eric W. "Abi-Khuzam Inequality." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Abi-KhuzamInequality.html

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