In mathematics, Nesbitt's inequality is a special case of the Shapiro inequality. It states that for positive real numbers a, b and c we have:
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Starting from Nesbitt's inequality(1903)
we transform the left hand side:
Now this can be transformed into:
Division by 3 and the right factor yields:
Now on the left we have the arithmetic mean and on the right the harmonic mean, so this inequality is true.
We might also want to try to use GM for three variables.
Suppose , we have that
define
The scalar product of the two sequences is maximum because of the Rearrangement inequality if they are arranged the same way, call and the vector shifted by one and by two, we have:
Addition yields Nesbitt's inequality.
The following identity is true for all
This clearly proves that the left side is no less than for positive a,b and c.
Note: every rational inequality can be solved by transforming it to the appropriate identity, see Hilbert's seventeenth problem.
Starting from Nesbitt's inequality(1903)
We add to both sides.
Now this can be transformed into:
Multiply by on both sides.
Which is true by the Cauchy-Schwarz inequality.
Starting from Nesbitt's inequality (1903)
we substitute a+b=x, b+c=y, c+a=z.
Now, we get
this can be transformed to
which is true, by inequality of arithmetic and geometric means.
This article incorporates material from Nesbitt's inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from proof of Nesbitt's inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.