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on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who attribute it to an unpublished result by Ky Fan.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 0 ≤ x i ≤ 1 2 {\textstyle 0\leq x_{i}\leq {\frac {1}{2}}} for i = 1, ..., n, then
( ∏ i = 1 n x i ) 1 / n ( ∏ i = 1 n ( 1 − x i ) ) 1 / n ≤ 1 n ∑ i = 1 n x i 1 n ∑ i = 1 n ( 1 − x i ) {\displaystyle {\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 = ⋅ ⋅ ⋅ = xn.
Remark Let
A n := 1 n ∑ i = 1 n x i , G n = ( ∏ i = 1 n x i ) 1 / n {\displaystyle 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, . . ., xn, and let
A n ′ := 1 n ∑ i = 1 n ( 1 − x i ) , G n ′ = ( ∏ i = 1 n ( 1 − x i ) ) 1 / n {\displaystyle 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 − xn. Then the Ky Fan inequality can be written as
G n G n ′ ≤ A n A n ′ , {\displaystyle {\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 Gn ≤ An.
Generalization with weights If xi ∈ [0,1/2] and γi ∈ [0,1] for i = 1, . . ., n are real numbers satisfying γ1 + . . . + γn = 1, then
∏ i = 1 n x i γ i ∏ i = 1 n ( 1 − x i ) γ i ≤ ∑ i = 1 n γ i x i ∑ i = 1 n γ i ( 1 − x i ) {\displaystyle {\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
γixi = 0 for all i = 1, . . ., n or all xi > 0 and there exists x ∈ (0,1/2] such that x = xi for all i = 1, . . ., n with γi > 0. The classical version corresponds to γi = 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 x 1 − x , x ∈ ( 0 , 1 2 ] . {\displaystyle 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 xi 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 γixi = 0 for all i = 1, . . ., n. (b) Assume now that all xi > 0. If there is an i with γi = 0, then the corresponding xi > 0 has no effect on either side of the inequality, hence the ith term can be omitted. Therefore, we may assume that γi > 0 for all i in the following. If x1 = x2 = . . . = xn, then equality holds. It remains to show strict inequality if not all xi are equal. The function f is strictly concave on (0,1/2], because we have for its second derivative
f ″ ( x ) = − 1 x 2 + 1 ( 1 − x ) 2 < 0 , x ∈ ( 0 , 1 2 ) . {\displaystyle 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
ln ∏ i = 1 n x i γ i ∏ i = 1 n ( 1 − x i ) γ i = ln ∏ i = 1 n ( x i 1 − x i ) γ i = ∑ i = 1 n γ i f ( x i ) < f ( ∑ i = 1 n γ i x i ) = ln ∑ i = 1 n γ i x i ∑ i = 1 n γ i ( 1 − x i ) , {\displaystyle {\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 γi 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