Nesbitt's inequality

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:

Proof

First proof: AM-HM inequality

By the AM-HM inequality on ,

Clearing denominators yields

from which we obtain

by expanding the product and collecting like denominators. This then simplifies directly to the final result.

Second proof: Rearrangement

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.

Third proof: Hilbert's Seventeenth Problem

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.

Fourth proof: Cauchy–Schwarz

Invoking the Cauchy–Schwarz inequality on the vectors yields

which can be transformed into the final result as we did in the AM-HM proof.

Fifth proof: AM-GM

We first employ a Ravi substitution: let . We then apply the AM-GM inequality to the set of six values to obtain

Dividing by yields

Substituting out the in favor of yields

which then simplifies to the final result.

Sixth proof: Titu's Screw lemma

Titu's lemma, a direct consequence of the Cauchy–Schwarz inequality, states that for any sequence of real numbers and any sequence of positive numbers , . We use its three-term instance with -sequence and -sequence :

By multiplying out all the products on the lesser side and collecting like terms, we obtain

which simplifies to

By the rearrangement inequality, we have , so the fraction on the lesser side must be at least . Thus,

Seventh proof: Homogeneous

As the left side of the inequality is homogeneous, we may assume . Now define , , and . The desired inequality turns into , or, equivalently, . This is clearly true by Titu's Lemma.

Eighth proof: Jensen inequality

Define and consider the function . This funcion can be shown to be convex in and, invoking Jensen inequality, we get

A straightforward computation yields

References

  • Nesbitt, A.M., Problem 15114, Educational Times, 3(2), 1903.
  • Ion Ionescu, Romanian Mathematical Gazette, Volume XXXII (September 15, 1926 - August 15, 1927), page 120
  • Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.
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