Ky Fan inequality

For Ky Fan inequality in game theory, see Ky Fan inequality (game theory).

In mathematics, the term Ky Fan inequality refers to an inequality involving the geometric mean and arithmetic mean of two sets of real numbers within the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who attribute it to an unpublished result by Ky Fan. They discuss the inequality in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of that inequality via forward-backward induction—a method that can also be used to prove the Ky Fan inequality.

This inequality is a special case of Levinson's inequality and serves as a foundation for several generalizations and refinements, some of which are referenced below.

Statement of the classical version

If with ${\textstyle 0\leq x_{i}\leq {\frac {1}{2}}}$ for i = 1, ..., n, then

${\frac {{\bigl (}\prod _{i=1}^{n}x_{i}{\bigr )}^{1/n}}{{\bigl (}\prod _{i=1}^{n}(1-x_{i}){\bigr )}^{1/n}}}\leq {\frac {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}}{{\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i})}}$

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

Remark

Let

$A_{n}:={\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad G_{n}={\biggl (}\prod _{i=1}^{n}x_{i}{\biggr )}^{1/n}$

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

$A_{n}':={\frac {1}{n}}\sum _{i=1}^{n}(1-x_{i}),\qquad G_{n}'={\biggl (}\prod _{i=1}^{n}(1-x_{i}){\biggr )}^{1/n}$

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

${\frac {G_{n}}{G_{n}'}}\leq {\frac {A_{n}}{A_{n}'}},$

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

Generalization with weights

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

${\frac {\prod _{i=1}^{n}x_{i}^{\gamma _{i}}}{\prod _{i=1}^{n}(1-x_{i})^{\gamma _{i}}}}\leq {\frac {\sum _{i=1}^{n}\gamma _{i}x_{i}}{\sum _{i=1}^{n}\gamma _{i}(1-x_{i})}}$

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 ∈ (0,⁠1/2⁠] 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.

Proof of the generalization

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

$f(x):=\ln x-\ln(1-x)=\ln {\frac {x}{1-x}},\qquad x\in (0,{\tfrac {1}{2}}].$

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 (0,⁠1/2⁠], because we have for its second derivative

$f''(x)=-{\frac {1}{x^{2}}}+{\frac {1}{(1-x)^{2}}}<0,\qquad x\in (0,{\tfrac {1}{2}}).$

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

${\begin{aligned}\ln {\frac {\prod _{i=1}^{n}x_{i}^{\gamma _{i}}}{\prod _{i=1}^{n}(1-x_{i})^{\gamma _{i}}}}&=\ln \prod _{i=1}^{n}{\Bigl (}{\frac {x_{i}}{1-x_{i}}}{\Bigr )}^{\gamma _{i}}\\&=\sum _{i=1}^{n}\gamma _{i}f(x_{i})\\&<f{\biggl (}\sum _{i=1}^{n}\gamma _{i}x_{i}{\biggr )}\\&=\ln {\frac {\sum _{i=1}^{n}\gamma _{i}x_{i}}{\sum _{i=1}^{n}\gamma _{i}(1-x_{i})}},\end{aligned}}$

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.

References

• Alzer, Horst (1988). "Verschärfung einer Ungleichung von Ky Fan". Aequationes Mathematicae. 36 (2–3): 246–250. doi:10.1007/BF01836094. MR 0972289. S2CID 122304838.

• Beckenbach, Edwin Ford; Bellman, Richard Ernest (1961). Inequalities. Berlin–Göttingen–Heidelberg: Springer-Verlag. ISBN 978-3-7643-0972-5. MR 0158038. {{cite book}}: ISBN / Date incompatibility (help)

• Moslehian, M. S. (2011). "Ky Fan inequalities". Linear and Multilinear Algebra. to appear. arXiv:1108.1467. Bibcode:2011arXiv1108.1467S.

• Neuman, Edward; Sándor, József (2002). "On the Ky Fan inequality and related inequalities I" (PDF). Mathematical Inequalities & Applications. 5 (1): 49–56. doi:10.7153/mia-05-06. MR 1880271.

• Neuman, Edward; Sándor, József (August 2005). "On the Ky Fan inequality and related inequalities II" (PDF). Bulletin of the Australian Mathematical Society. 72 (1): 87–107. doi:10.1017/S0004972700034894. MR 2162296. Archived from the original (PDF) on 2020-03-25. Retrieved 2007-06-14.

• Sándor, József; Trif, Tiberiu (1999). "A new refinement of the Ky Fan inequality" (PDF). Mathematical Inequalities & Applications. 2 (4): 529–533. doi:10.7153/mia-02-43. MR 1717045.

External links

• Ky Fan at the Mathematics Genealogy Project