几个著名的不等式Some famous inequalities

88° Some famous inequalities.

Since the time of Pythagoras, the Greeks were familiar with three means, called the arithmetic, the geometric, and the subcontrary—the last name being later changed to harmonic by Archytas and Hippasus. We may define these three means of two positive numbers a and b as

A=(a+b)/2, G=√ab, H=2ab/(a+b),

respectively. A favorite question, often asked on a Master's oral examination, is to show that A ≥ G ≥ H, with equality if and only if a = b. A singularly neat establishment of these inequalities appears in Book III of Pappus's Mathematical Collection. Take B on segment AC (see Figure 12), and erect the perpendicular to AC at B to cut the semicircle on AC in D. Then, if F is the foot of the perpendicular from B on OD, where O is the midpoint of AC, one may easily show that OD, BD, FD represent the arithmetic mean, the geometric mean, and the harmonic mean of the segments AB and BC. [That OD is the arithmetic mean is obvious. That BD is the geometric mean is well known from high school geometry. That FD is the harmonic mean follows from the similar triangles DFB and DBO, for we have FD/DB = DB/OD, whence FD = (DB)²/OD = 2(AB)(BC)/(AB + BC).] The concerned inequalities are now geometrically evident. The Master's candidate asked to establish these inequalities could hardly do better than to present the Pappus argument, and he would surely make a more impressive showing than by stumbling about with purely algebraic procedures.