Geometric Mean

$$G\equiv \left({\,\prod_{i=1}^n a_i}\right)^{1/n}.$$

Hoehn and Niven (1985) show that

$$G(a_1+c, a_2+c, \ldots, a_n+c)>c+G(a_1, a_2, \ldots, a_n)$$

for any Positive constant $c$.

See also Arithmetic Mean, Arithmetic-Geometric Mean, Carleman's Inequality, Harmonic Mean, Mean, Root-Mean-Square

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Hoehn, L. and Niven, I. ``Averages on the Move.'' Math. Mag. 58, 151-156, 1985.