Ky Fan inequality

If with $0\backslash le\; x\_i\backslash le\; \backslash frac\{1\}\{2\}$ for i = 1, ..., n, then

:$\backslash frac\{\; \backslash bigl(\backslash prod\_\{i=1\}^n\; x\_i\backslash bigr)^\{1/n\}\; \}$

{ \bigl(\prod_{i=1}^n (1-x_i)\bigr)^{1/n} }

\le

\frac{ \frac1n \sum_{i=1}^n x_i }

{ \frac1n \sum_{i=1}^n (1-x_i) }

with equality if and only if x₁ = x₂ = ⋅ ⋅ ⋅ = x_n.

Let

:$A\_n:=\backslash frac1n\backslash sum\_\{i=1\}^n\; x\_i,\backslash qquad\; G\_n=\backslash biggl(\backslash prod\_\{i=1\}^n\; x\_i\backslash biggr)^\{1/n\}$

denote the arithmetic and geometric mean, respectively, of x₁, . . ., x_n, and let

:$A\_n\text{'}:=\backslash frac1n\backslash sum\_\{i=1\}^n\; (1-x\_i),\backslash qquad\; G\_n\text{'}=\backslash biggl(\backslash prod\_\{i=1\}^n\; (1-x\_i)\backslash biggr)^\{1/n\}$

denote the arithmetic and geometric mean, respectively, of 1 − x₁, . . ., 1 − x_n. Then the Ky Fan inequality can be written as

:$\backslash frac\{G\_n\}\{G\_n\text{'}\}\backslash le\backslash frac\{A\_n\}\{A\_n\text{'}\},$

which shows the similarity to the inequality of arithmetic and geometric means given by G_n ≤ A_n.

If x_i ∈ [0,{{sfrac|1|2}}] and γ_i ∈ [0,1] for i = 1, . . ., n are real numbers satisfying γ₁ + . . . + γ_n = 1, then

:$\backslash frac\{\; \backslash prod\_\{i=1\}^n\; x\_i^\{\backslash gamma\_i\}\; \}$

{ \prod_{i=1}^n (1-x_i)^{\gamma_i} }

\le

\frac{ \sum_{i=1}^n \gamma_i x_i }

{ \sum_{i=1}^n \gamma_i (1-x_i) }

with the convention 0⁰ := 0. Equality holds if and only if either

• γ_ix_i = 0 for all i = 1, . . ., n or
• all x_i > 0 and there exists x ∈ (0,{{sfrac|1|2}}] such that x = x_i for all i = 1, . . ., n with γ_i > 0.

The classical version corresponds to γ_i = 1/n for all i = 1, . . ., n.

Idea: Apply Jensen's inequality to the strictly concave function

:$f(x):=\; \backslash ln\; x-\backslash ln(1-x)\; =\; \backslash ln\backslash frac\; x\{1-x\},\backslash qquad\; x\backslash in(0,\backslash tfrac12].$

Detailed proof: (a) If at least one x_i is zero, then the left-hand side of the Ky Fan inequality is zero and the inequality is proved. Equality holds if and only if the right-hand side is also zero, which is the case when γ_ix_i = 0 for all i = 1, . . ., n.

(b) Assume now that all x_i > 0. If there is an i with γ_i = 0, then the corresponding x_i > 0 has no effect on either side of the inequality, hence the i^th term can be omitted. Therefore, we may assume that γ_i > 0 for all i in the following. If x₁ = x₂ = . . . = x_n, then equality holds. It remains to show strict inequality if not all x_i are equal.

The function f is strictly concave on (0,{{sfrac|1|2}}], because we have for its second derivative

:

Using the functional equation for the natural logarithm and Jensen's inequality for the strictly concave f, we obtain that

:$$

\begin{align}

\ln\frac{ \prod_{i=1}^n x_i^{\gamma_i}}

{ \prod_{i=1}^n (1-x_i)^{\gamma_i} }

&=\ln\prod_{i=1}^n\Bigl(\frac{x_i}{1-x_i}\Bigr)^{\gamma_i}\\

&=\sum_{i=1}^n \gamma_i f(x_i)\\

&

&=\ln\frac{ \sum_{i=1}^n \gamma_i x_i }

{ \sum_{i=1}^n \gamma_i (1-x_i) },

\end{align}

where we used in the last step that the γ_i sum to one. Taking the exponential of both sides gives the Ky Fan inequality.

• {{cite journal

|last = Alzer

|first = Horst

|title = Verschärfung einer Ungleichung von Ky Fan

|journal = Aequationes Mathematicae

|volume = 36

|issue = 2–3

|pages = 246–250

|year = 1988

|url = https://link.springer.com/article/10.1007/BF01836094

|mr = 972289

|doi = 10.1007/BF01836094

|s2cid = 122304838

|url-access= subscription

}}

• {{cite book

| last = Beckenbach

| first = Edwin Ford

|author2=Bellman, Richard Ernest |authorlink2=Richard E. Bellman

| title = Inequalities

| publisher = Springer-Verlag

| year = 1961

| location = Berlin–Göttingen–Heidelberg

|mr=158038

| isbn = 978-3-7643-0972-5

}}

• {{cite journal

| last = Moslehian

| first = M. S.

| title = Ky Fan inequalities

| journal = Linear and Multilinear Algebra

| volume = to appear

| arxiv = 1108.1467| bibcode = 2011arXiv1108.1467S

| year = 2011

}}

• {{cite journal

| last = Neuman

| first = Edward

|author2=Sándor, József

| title = On the Ky Fan inequality and related inequalities I

| journal = Mathematical Inequalities & Applications

| volume = 5

| issue = 1

| pages = 49–56

| year = 2002

| url = http://www.ele-math.com/files/mia/05-1/full/mia-05-06.pdf

|mr=1880271

| doi = 10.7153/mia-05-06

| doi-access = free

}}

• {{cite journal

| last = Neuman

| first = Edward

|author2=Sándor, József

| title = On the Ky Fan inequality and related inequalities II

| journal = Bulletin of the Australian Mathematical Society

| volume = 72

| issue = 1

| pages = 87–107

|date=August 2005

| url = http://www.austms.org.au/Publ/Bulletin/V72P1/pdf/721-5068-NeSa.pdf

|mr=2162296

| doi = 10.1017/S0004972700034894

| doi-access = free

}}

• {{cite journal

| last = Sándor

| first = József

|author2=Trif, Tiberiu

| title = A new refinement of the Ky Fan inequality

| journal = Mathematical Inequalities & Applications

| volume = 2

| issue = 4

| pages = 529–533

| year = 1999

| url = http://www.ele-math.com/files/mia/02-4/full/mia-02-43.pdf

|mr=1717045

| doi = 10.7153/mia-02-43

| doi-access = free

}}

• {{Mathgenealogy|name = Ky Fan|id = 15631}}

Category:Inequalities (mathematics)

Category:Articles containing proofs