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Abi-Khuzam Inequality

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The Abi-Khuzam inequality states that

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

where A, B, and C are the vertex angles of a triangle, measured in radians. The maximum of sinAsinBsinC/(ABC) is attained iff the triangle is an equilateral triangle (and therefore A=B=C=pi/3). This maximum equals [3sqrt(3)/(2pi)]^3=81sqrt(3)/(8pi^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 sines 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. 84, 53-62, 1978. https://doi.org/10.1090/S0002-9904-1978-14405-X.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 Resource. https://mathworld.wolfram.com/Abi-KhuzamInequality.html

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