The Ladder of Power Means Các định nghĩa thích hợp của sức mạnh có nghĩa là M0 được thúc đẩy bởi mong muốn tự nhiên để làm cho bản đồ t → Mt một chức năng liên tục trên tất cả các của R. vấn đề thách thức đầu tiên cho thấy làm thế nào điều này có thể đạt được, và nó cũng cho biết thêm một lớp mới của trực giác sự hiểu biết của chúng ta về có nghĩa là hình học. | 8 The Ladder of Power Means The quantities that provide the upper bound in Cauchy s inequality are special cases of the general means f n I 1 t Mt Mt x p 02Pkxk f k 1 where p p1 p2 . pn is a vector of positive weights with total mass of Pl p2 pn 1 and x x1 x2 . xn is a vector of nonnegative real numbers. Here the parameter t can be taken to be any real value and one can even take t TO or t TO although in these cases and the case t 0 the general formula requires some reinterpretation. The proper definition of the power mean Mo is motivated by the natural desire to make the map t Mt a continuous function on all of R. The first challenge problem suggests how this can be achieved and it also adds a new layer of intuition to our understanding of the geometric mean. Problem The Geometric Mean as a Limit For nonnegative real numbers xk k 1 2 . . n and nonnegative weights Pk k 1 2 . . n with total mass Pl P2 pn 1 one has the limit Approximate Equalities and Landau s Notation The solution of this challenge problem is explained most simply with the help of Landau s little o and big O notation. In this useful shorthand the statement limt 0 f t g t 0 is abbreviated simply by writing 120 The Ladder of Power Means 121 f t o g t as t 0 and analogously the statement that the ratio f t g t is bounded in some neighborhood of 0 is abbreviated by writing f t O g t as t 0. By hiding details that are irrelevant this notation often allows one to render a mathematical inequality in a form that gets most quickly to its essential message. For example it is easy to check that for all x 1 one has a natural two-sided estimate for log 1 x ỉ edu log 1 x x 1 x J1 u yet for many purposes these bounds are more efficiently summarized by the simpler statement log 1 x x O x2 as x 0. Similarly one can check that for all x 1 one has the bound XX x 2 xj - 2 1 x e 1 x x2 1 x ex2 though again for many calculations we only need to know that these bounds give us the relation ex 1 x O x2 .
