tailieunhanh - THE CAUCHY – SCHWARZ MASTER CLASS - PART 5

Consequences of Order Một trong những câu hỏi tự nhiên đi kèm với bất kỳ sự bất bình đẳng là khả năng thừa nhận một trò chuyện của một loại hay cách khác. Khi chúng tôi đặt ra câu hỏi này cho bất đẳng thức Cauchy, chúng tôi tìm thấy một vấn đề thách thức đó là chắc chắn giá trị sự chú ý của chúng tôi. Nó không chỉ dẫn đến kết quả là hữu ích trong quyền riêng của họ, nhưng nó cũng đặt chúng ta trên con đường của một trong những nguyên tắc cơ bản nhất. | 5 Consequences of Order One of the natural questions that accompanies any inequality is the possibility that it admits a converse of one sort or another. When we pose this question for Cauchy s inequality we find a challenge problem that is definitely worth our attention. It not only leads to results that are useful in their own right but it also puts us on the path of one of the most fundamental principles in the theory of inequalities the systematic exploitation of order relationships. Problem The Hunt for a Cauchy Converse Determine the circumstances which suffice for nonnegative real numbers ak bk k 1 2 . . n to satisfy an inequality of the type 1 2 n Ẽ k 1 ak n p ffikbk k i for a given constant p. Orientation Part of the challenge here is that the problem is not fully framed there are circumstances and conditions that remain to be determined. Nevertheless uncertainty is an inevitable part of research and practice with modestly ambiguous problems can be particularly valuable. In such situations one almost always begins with some experimentation and since the case n 1 is trivial the simplest case worth study is given by taking the vectors 1 a and 1 b with a 0 and b 0. In this case the two sides of the conjectured Cauchy converse relate the quantities 1 a2 1 1 b2 1 and 1 ab 73 74 Consequences of Order and this calculation already suggests a useful inference. If a and b are chosen so that the product ab is held constant while a TO then one finds that the right-hand expression is bounded but the left-hand expression is unbounded. This observation shows in essence that for a given fixed value of p 1 the conjecture cannot hold unless the ratios ak bk are required to be bounded from above and below. Thus we come to a more refined point of view and we see that it is natural to conjecture that a bound of the type will hold provided that the summands satisfy the ratio constraint m M for all k 1 2 . n for some constants 0 m M TO. In this new .