The practical usage of weighted mean in cotton had been expounded through calculated method of circular cone position in cotton bolls and mean of boll weight,and the practical usage of geometric mean,the growth per cent of plant height a day had been expounded through analysing exponential increasing phase of plant height in cotton.

A new functional inequality was established as one of the two applications,which could prove the notable inequality that the geometrical average is no more than the algebraic average,and could also prove its related forms.

In this paper, the relation X≥ X G≥ X H between arithmetic average X, geometrical average  X G and harmonic average  X H of positive number X i>0 (i=1,2,…,n) is proved.

Two lemmas of trigonometric inequality can be deduced by the application of the principle that arithmetic average is greater than or equal to geometric average.

In order to emphasize the influence of over-irrigation or insufficient irrigation on irrigation uniformity,this paper put for- ward the concept of distribution uniformity coefficient based on geometric average,which is defined as the ratio between geometric average of the depth of partial measuring points and arithmetic average of all points.

Discrete weighted geometric average inequality;