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THE CAUCHY – SCHWARZ MASTER CLASS - PART 7

Integral Intermezzo Sự bất bình đẳng cơ bản nhất là cho các khoản tiền hữu hạn, nhưng có thể không có nghi ngờ rằng sự bất bình đẳng cho tích phân cũng xứng đáng được chia sẻ công bằng của sự chú ý của chúng tôi. Tích phổ biến trên toàn khoa học và kỹ thuật, và họ cũng có một số ưu điểm toán học trên số tiền. Ví dụ, tích phân có thể được cắt thành miếng như nhiều như chúng tôi muốn, và hội nhập của các bộ phận là hầu như luôn luôn duyên dáng hơn. | 7 Integral Intermezzo The most fundamental inequalities are those for finite sums but there can be no doubt that inequalities for integrals also deserve a fair share of our attention. Integrals are pervasive throughout science and engineering and they also have some mathematical advantages over sums. For example integrals can be cut up into as many pieces as we like and integration by parts is almost always more graceful than summation by parts. Moreover any integral may be reshaped into countless alternative forms by applying the change-of-variables formula. Each of these themes contributes to the theory of integral inequalities. These themes are also well illustrated by our favorite device concrete challenge problems which have a personality of their own. Problem 7.1 A Continuum of Compromise Show that for an integrable f R R one has the bound l- TO I- TO 1 TO 1 I If x dx 8 1f I xf x 2 d j f I If x 2 dxj . 7.1 J TO J TO J TO A Quick Orientation and a Qualitative Plan The one-fourth powers on the right side may seem strange but they are made more reasonable if one notes that each side of the inequality is homogenous of order one in f that is if f is replaced by Af where A is a positive constant then each side is multiplied by A. This observation makes the inequality somewhat less strange but one may still be stuck for a good idea. We faced such a predicament earlier where we found that one often does well to first consider a simpler qualitative challenge. Here the nat 105 106 Integral Intermezzo ural candidate is to try to show that the left side is finite whenever both integrals on the right are finite. Once we ask this question we are not likely to need long to think of looking for separate bounds for the integral of f x on the interval T t t and its complement Tc. If we also ask ourselves how we might introduce the term xf x then we are almost forced to think of using the splitting trick on the set Tc . Pursuing this thought we then find for all t 0 that we .