After taking reciprocals of the left‐ and right‐hand sides, Eq. (2.53) gives rise to
(2.54)
or, after recalling Eq. (2.43),
Therefore, the harmonic mean of two numbers never exceeds the geometric mean (again encompassing only positive values), and coincides therewith again when x1 = x2.
When n = 2, another mean can be defined as
– known as logarithmic mean; Eq. (2.56) is often rephrased to
after taking advantage of Eq. (2.26). This logarithmic mean lies below the arithmetic and the geometric means, i.e.
where Eq. (2.55) was meanwhile taken advantage of. To prove so, it is convenient to insert Eqs. (2.42), (2.43), (2.45), and (2.57) pertaining to x1 and x2 so as to get
(2.59)
from Eq. (2.58), where factoring of x1 ≥ x2 > 0 (as per working hypothesis) in all sides gives rise to
after dropping x1 > 0 from all sides, and further multiplying and dividing the last side by x2/x1, Eq. (2.60) turns to
– where z denotes an auxiliary variable satisfying
A graphical account of Eq. (2.60) is provided in Fig. 2.3. Inspection of the curves therein not only unfolds a clear and systematic positioning of the various means relative to each other – in general agreement with (so far, postulated) Eq. (2.58) – but also indicates a collective convergence to x1 as x2 approaches it (as expected).
Figure 2.3 Variation of arithmetic mean (arm), logarithmic mean (lom), geometric mean (gem), and harmonic mean (ham) of positive x1 and x2 < x1, normalized by x1, as a function of their ratio, x2/x1.
To provide a quantitative argument in support of the graphical trends above, one may expand the middle left‐ and middle right‐hand sides of Eq. (2.61) via Taylor’s series (see discussion later), around z = 1, according to
where 1/2 in the left‐hand side was meanwhile splitted as 1 − 1/2; note that such series in z are convergent because 0 < z < 1, as per Eq. (2.62). Straightforward simplification of Eq. (2.63) unfolds
(2.64)
where z − 1 may be further dropped off both numerator and denominator of the middle side to give
after simplifying notation in Eq. (2.65) to
(2.66)
with the aid of
(2.67)
one may add ζ/2 to all sides to get
– where all terms in the denominator of the middle side are positive, whereas all terms (besides 1) in the right‐hand side are negative. Long (polynomial) division of 1 by 1 + ζ/2 + ζ2/3 + ζ3/4 + ⋯ (according to an algorithm to be presented below) allows further transformation of Eq. (2.68) to
(2.69)
where condensation of terms alike in the middle side unfolds