Neuman–Sándor mean

In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as:

${\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}}$

This mean interpolates the inequality of the unweighted arithmetic mean A = (a + b)/2) and of the second Seiffert mean T defined as:

${\displaystyle T(a,b)={\frac {a-b}{2\arctan \left({\frac {a-b}{a+b}}\right)}},}$

so that A < M < T.

The M(a,b) mean, introduced by Edward Neuman and József Sándor,^[1] has recently been the subject of intensive research and many remarkable inequalities for this mean can be found in the literature.^[2] Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean.^{[3][4][5][6][7]} Neuman and others utilized this mean to study other bivariate means and inequalities.^{[8][9][10][11][12]}