Let $p$ be a real number. Lehmer mean of the positive numbers $a_1,\,\ldots,\,a_n$ is defined as

(1)

The Lehmer mean of certain numbers is the greater the greater is the parametre $p$ , i.e. $$L_p(a_1,\,\ldots,\,a_n) \;\geqq\; L_q(a_1,\,\ldots,\,a_n) \quad \forall\; p \;>\; q.$$ This turns out from the nonnegativeness of the partial derivative of $L_p$ with respect to $p$ ; in the case $n =2$ it writes $$\frac{\partial L_p}{\partial p} \;=\; \frac{a^{p-1}b^{p-1}(a\!-\!b)(\ln{a}-\ln{b})}{(a^{p-1}\!+\!b^{p-1})^2} \;\geqq\; 0.$$ Thus in the below list containing special cases of Lehmer mean, the harmonic mean is the least and the contraharmonic the greatest (cf. the comparison of Pythagorean means).

E.g. for two arguments $a$ and $b$ , one has

• $\displaystyle L_0(a,\,b) \,=\, \frac{2ab}{a\!+\!b}$ ,     harmonic mean,
• $\displaystyle L_{1/2}(a,\,b) \,=\, \sqrt{ab}$ ,     geometric mean,
• $\displaystyle L_1(a,\,b) \,=\, \frac{a\!+\!b}{2}$ ,     arithmetic mean,
• $\displaystyle L_2(a,\,b) \,=\, \frac{a^2\!+\!b^2}{a\!+\!b}$ ,     contraharmonic mean.

Note. The least and the greatest of the numbers $a_1,\,\ldots,\,a_n$ may be regarded as borderline cases of the Lehmer mean, since $$\lim_{p\to-\infty}L_p(a_1,\,\ldots,\,a_n) \;=\; \min\{a_1,\,\ldots,\,a_n\}, \quad \lim_{p\to+\infty}L_p(a_1,\,\ldots,\,a_n) \;=\; \max\{a_1,\,\ldots,\,a_n\}.$$ For proving these equations, suppose that there are exactly $k$ greatest (resp. least) ones among the numbers and that those are $a_1 = \ldots = a_k$ . Then we can write $$L_p(a_1,\,\ldots,\,a_n) \;=\; \frac{a_1^p\left[k+\!\left(\frac{a_{k+1}}{a_1}\right)^p\!+\ldots+\!\left(\frac{a_{n}}{a_1}\right)^p\right]} {a_1^{p-1}\left[k+\!\left(\frac{a_{k+1}}{a_1}\right)^{p-1}\!+\ldots+\!\left(\frac{a_{n}}{a_1}\right)^{p-1}\right]}.$$ Letting $p \to +\infty$ (resp. $p \to -\infty$ ), this equation yields $$L_p(a_1,\,\ldots,\,a_n) \;\longrightarrow\; a_1.$$