A harmonic mean inequality for the q-gamma function

In 1984, Kairies proved that for all positive real numbers x the geometric mean of Gamma(q)(x) and Gamma(q)(1/x) is greater than or equal to 1, that is, 1 <= root Gamma(q)(x)Gamma(q)(1/x) (0 < q not equal 1), where Gamma(q) denotes the q-gamma function. This result can be improved if q is an element of (0, 1). We show that for all x > 0 the harmonic mean of Gamma(q) (x) and Gamma(q) (1/x) is greater than or equal to 1, that is, 1 <= 2/1/ Gamma(q) (x) + 1/ Gamma(q) (1/x) (0 < q < 1) with equality if and only if x = 1.

Automated summary

The study by Kairies in 1984 showed that for positive real numbers x, the geometric mean of Gamma(q)(x) and Gamma(q)(1/x) is greater than or equal to 1 if q is not equal to 1. This result can be further improved if q is an element of (0,1).