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Heronian mean is between geometric and arithmetic mean
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(Theorem)
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Theorem. For non-negative numbers $x$ and $y$ , the inequalities $$\sqrt{xy} \;\leqq\; \frac{x\!+\!\sqrt{xy}\!+\!y}{3} \;\leqq\; \frac{x\!+\!y}{2}$$ are in force, i.e. the Heronian mean is always at least equal to the geometric mean and at most equal to the arithmetic mean. The equality signs are true if and only if $x = y$ .
Proof.
$1^\circ.$
$2^\circ.$
All consecutive inequalities of both chains are equivalent since $x$ and $y$ are non-negative. As for the equalities, the chains are valid with the mere equality signs.
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"Heronian mean is between geometric and arithmetic mean" is owned by pahio.
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Cross-references: proof, equality, arithmetic mean, geometric mean, Heronian mean, inequalities, numbers, theorem
There is 1 reference to this entry.
This is version 7 of Heronian mean is between geometric and arithmetic mean, born on 2008-02-17, modified 2008-03-17.
Object id is 10284, canonical name is HeronianMeanIsBetweenGeometricAndArithmeticMean.
Accessed 1170 times total.
Classification:
| AMS MSC: | 00A05 (General :: General and miscellaneous specific topics :: General mathematics) | | | 01A20 (History and biography :: History of mathematics and mathematicians :: Greek, Roman) | | | 26B99 (Real functions :: Functions of several variables :: Miscellaneous) | | | 26D07 (Real functions :: Inequalities :: Inequalities involving other types of functions) |
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Pending Errata and Addenda
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