Ky Fan Inequality - Generalization With Weights

Generalization With Weights

If xi ∈ and γi ∈ for i = 1, . . ., n are real numbers satisfying γ1 + . . . + γn = 1, then

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

  • γixi = 0 for all i = 1, . . ., n or
  • all xi > 0 and there exists x ∈ (0,½] 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.

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