tailieunhanh - THE CAUCHY – SCHWARZ MASTER CLASS - PART 9

H¨lder’s Inequality o Bốn kết quả cung cấp lõi trung tâm của lý thuyết cổ điển của sự bất bình đẳng, và chúng tôi đã nhìn thấy ba trong số này: bất đẳng thức Cauchy-Schwarz bất bình đẳng AM-GM, và bất bình đẳng của Jensen. Tứ tấu được hoàn thành bởi một kết quả lần đầu tiên thu được bằng LC Rogers năm 1888 và bắt nguồn một cách khác một năm sau đó bởi Otto H ¨ lder. | 9 Holder s Inequality Four results provide the central core of the classical theory of inequalities and we have already seen three of these the Cauchy-Schwarz inequality the AM-GM inequality and Jensen s inequality. The quartet is completed by a result which was first obtained by . Rogers in 1888 and which was derived in another way a year later by Otto Holder. Cast in its modern form the inequality asserts that for all nonnegative ak and bk k 1 2 . n one has the bound id abk Ề apy 7f bq 1 q k 1 k 1 k 1 provided that the powers p 1 and q 1 satisfy the relation p q 1. Ironically the articles by Rogers and Holder leave the impression that these authors were mainly concerned with the extension and application of the AM-GM inequality. In particular they did not seem to view their version of the bound as singularly important though Rogers did value it enough to provide two proofs. Instead the opportunity fell to Frigyes Riesz to cast the inequality in its modern form and to recognize its fundamental role. Thus one can argue that the bound might better be called Rogers s inequality or perhaps even the Rogers-Holder-Riesz inequality. Nevertheless long ago the moving hand of history began to write Holder s inequality and now for one to use another name would be impractical though from time to time some acknowledgment of the historical record seems appropriate. The first challenge problem is easy to anticipate one must prove the inequality and one must determine the circumstances where equal 135 136 Holder s Inequality ity can hold. As usual readers who already know a proof of Holder s inequality are invited to discover a new one. Although new proofs of Holder s inequality appear less often than those for the Cauchy-Schwarz inequality or the AM-GM inequality one can have confidence that they can be found. Problem Holder s Inequality First prove Riesz s version of the inequality of Rogers 1888 and Holder 1889 then prove that one has .