In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Nesbitt's
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
states that for positive real numbers ''a'', ''b'' and ''c'',
:
It is an elementary special case (N = 3) of the difficult and much studied
Shapiro inequality In mathematics, the Shapiro inequality is an inequality (mathematics), inequality proposed by Harold S. Shapiro in 1954.
Statement of the inequality
Suppose n is a natural number and x_1, x_2, \dots, x_n are positive numbers and:
* n is even and ...
, and was published at least 50 years earlier.
There is no corresponding upper bound as any of the 3 fractions in the inequality can be made arbitrarily large.
Proof
First proof: AM-HM inequality
By the
AM-
HM inequality on
,
:
Clearing denominators In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.
Example
Co ...
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 In mathematics, the rearrangement inequality states that
x_n y_1 + \cdots + x_1 y_n
\leq x_ y_1 + \cdots + x_ y_n
\leq x_1 y_1 + \cdots + x_n y_n
for every choice of real numbers
x_1 \leq \cdots \leq x_n \quad \text \quad y_1 \leq \cdots \leq y_n
...
if they are arranged the same way, call
and
the vector
shifted by one and by two, we have:
:
:
Addition yields our desired Nesbitt's inequality.
Third proof: Sum of Squares
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 demonstrated by transforming it to the appropriate sum-of-squares identity, see
Hilbert's seventeenth problem
Hilbert's seventeenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It concerns the expression of positive definite rational functions as sums of quotients of squares. The original quest ...
.
Fourth proof: Cauchy–Schwarz
Invoking the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
on the vectors
yields
:
which can be transformed into the final result as we did in
the AM-HM proof.
Fifth proof: AM-GM
Let
. We then apply the
AM-GM inequality
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; a ...
to obtain the following
:
because
Substituting out the
in favor of
yields
:
:
which then simplifies to the final result.
Sixth proof: Titu's lemma
Titu's lemma, a direct consequence of the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality fo ...
, states that for any sequence of
real numbers
and any sequence of
positive numbers
,
.
We use the lemma on
and
. This gives,
:
This results in,
:
i.e.,
:
Seventh proof: Using homogeneity
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 function can be shown to be convex in