Generalized Means a.cyclohexane.molecule The Generalized Mean Consider the generalized mean for non-negative xi n !1=p 1 X M(p; x ; : : : ; x ) := xp 1 n n i i=1 whose name becomes self-explanatory when we consider the six cases n !−1 1 X −1 n M(−1; x1; : : : ; xn) = x = harmonic mean n i x−1 + ::: + x−1 i=1 1 n n !1=p 1 X p 1=n M(0; x1; : : : ; xn) = lim x = (x1 ··· xn) geometric mean p!0 n i i=1 n 1 X x1 + ::: + xn M(1; x ; : : : ; x ) = x = arithmetic mean 1 n n i n i=1 n 1=2 1 X x2 + ::: + x2 M(2; x ; : : : ; x ) = x2 = 1 n quadratic mean 1 n n i n i=1 n !1=p 1 X p M(−∞; x1; : : : ; xn) = lim x = minfx1; : : : ; xng minimum p→−∞ n i i=1 n !1=p 1 X p M(1; x1; : : : ; xn) = lim x = maxfx1; : : : ; xng maximum p!1 n i i=1 The three cases involving limits are somewhat more difficult to evaluate. In these cases, the calculations are simplified by first considering the natural logarithm of the limit and hence applying L'H^opital'srule. p p p p log((x1 + ::: + xn)=n) x1 log x1 + ::: + xn log xn log x1 ··· xn log L0 = lim = lim p p = p!0 p p!0 x1 + ::: + xn n p p p p 1 x1 + ::: + xn 1 1 x1 xn log L1 = lim log = log xk + lim log + ::: + 1 + ::: + = log xk p!1 p n p!1 p n xk xk 1 1 −1 1 1 −1 L−∞ = M 1; ;:::; = max ;:::; = min fx1; : : : ; xng x1 xn x1 xn p The first case is simple. In the second case we have chosen xk = maxfx1; : : : ; xng in order that all terms (xi=xk) , i 6= k, tend to 0 as p tends to 1; and we have sneakily shown the third case to be a special case of the second and thus have avoided needing to calculate a limit. Generalized Mean Inequalities From the six examples of the generalized mean above, it seems plausible that M monotonically increases with p. In order to prove this rigorously, we must show that the partial derivative of M with respect to p is non-negative for all p. As with the limits earlier, our calculations may be simplified if we instead consider log M. We obtain p p p p p p @ log M @ log(x1 + ::: + xn) − log n 1 x1 log x1 + ::: + xn log xn log x1 + ::: + log xn log n = = p p − 2 + 2 @p @p p p x1 + ::: + xn p p p p p p p p p Now normalizing the xi by substituting xi ! xi =(x1 + ::: + xn) results in x1 + ::: + xn = 1 and a simplification of our partial derivative to xp log x + ::: + xp log x log n 1 1 1 n n + = (xp log xp + ::: + xp log xp + log n) p p2 p2 1 1 n n In order to continue with our analysis, we will first prove Jensen's inequality. Generalized Means a.cyclohexane.molecule Theorem. For a convex function f with c1 + ··· + cn = 1, f(c1x1 + ··· + cnxn) ≤ c1f(x1) + ··· + cnf(xn). For a concave function, the inequality is reversed. Proof. Compare f and its tangent line g at c1x1 + ··· +cnxn. Then f(c1x1 + ··· +cnxn) = g(c1x1 + ··· +cnxn) = c1g(x1) + ··· + cng(xn) ≤ c1f(x1) + ··· + cnf(xn), where the first equality follows from the equivalence of the function and the tangent line at their point of tangency, the second from linearity, and the third from convexity. Remark. While a \linear" function g(x) = ax + b is not usually linear, the condition c1 + ··· + cn = 1 allows us to write g(c1x1 + ··· + cnxn) = a(c1x1 + ··· + cnxn) + 1b = a(c1x1 + ··· + cnxn) + (c1 + ··· + cn)b = c1ax + c1b + ··· + cnaxn + cnb = c1g(x1) + ··· + cng(xn), establishing linearity. p Apply Jensen's inequality on the function f(x) = x log x, x 2 (0; 1], letting ci = 1=n and f(xi) = xi for all i. Since f 00(x) = 1=x is convex on the interval (0; 1], we have the inequality 1 xp xp f(xp) f(xp ) f = f 1 + ::: + n ≤ 1 + ::: + n n n n n n 1 1 1 log ≤ (xp log xp + ::: + xp log xp ) n n n 1 1 n n p p p p − log n ≤ x1 log x1 + ::: + xn log xn p p p p 0 ≤ x1 log x1 + ::: + xn log xn + log n Hence, by our above expression for the partial derivative of M with respect to p, we have established that @M 1 = (xp log xp + ::: + xp log xp + log n) ≥ 0 @p p2 1 1 n n and that M is a monotonically increasing function. We have thus also established the inequalities min(x) ≤ HM(x) ≤ GM(x) ≤ AM(x) ≤ QM(x) ≤ max(x) for x := x1; : : : ; xn:.