Bất đẳng thức toán học và những viên kim cương: Phần 1

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Phần 1 Tài liệu Những viên kim cương trong bất đẳng thức toán học giới thiệu tới người đọc các nội dung: Những viên kim cương trong bất đẳng thức cổ điển, những viên kim cương trong bất đẳng thức cận đại, những viên kim cương trong giải tích. Đây là một Tài liệu hữu ích dành cho các bạn sinh viên và những ai đam mê toán học dùng làm Tài liệu học tập và nghiên cứu.

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Nội dung Text: Bất đẳng thức toán học và những viên kim cương: Phần 1

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TRAN PHU’O’NG C{)n_g téc vién: Trn Tuin Anh l\guyén Anh Cuimg; Bi1iVie_“:tAnh i NHU’NG VIEN KIM CU’O’NG TRONG BAT DANG THU’C TO/§\N HQC (Tdi bdn lcin thzi nhd’t) NHA XUi\T BAN TRI THU’C
MUC ugc CHU’O’NG I: NHU’NG VlEN KIM CU’O’NG TRONG BAT DANG THLPC CO DIEN 11 §1. Bit ding thirc AM - GM vi cic R9 thuit chqn aiém ro’i 12 §1.1. Bit ding thirc AM — GM 17 §1.2. NhCrng sic miu diim rdi trong bit ding thirc AM GM —~ 20 §2. Bit ding thirc Cauchy - Bunhiakowski — Schwarz vi ky 123 thuit chgn dié'm rdi §2.1. Bit ding thfrc Cauchy — Bunhiakowski Schwarz -— 123 Ky thuit chqn diim rdi trong bit ding thfrc Cauchy - ' ' BunhiaC6pski —Schwarz 126 §3. Bit ding thlfrc Holder vi ky thuit chgn diim ro’i 173 §3.1. Bit ding mac Holder 173 §3.2. Ky thuit su’ dung bit ding thllrc Holder 176 §4. Bit ding mac Minkowski vi kv thuit sir dung 193 §4.1. Bit ding thirc Minkowski 193 §4.2. Ky thuit su’ dung bit ding thirc Minkowski 196 §5. Bit ding thirc Chebyshev vi ky thuit sir dung 201 §s.1. Bit ding thirc Chebyshev 201 §5.2. Ky thuit sir dung bit ding thirc Chebyshev Z03 CHUUNG "3 Nl-‘|l~)’NG VIEN KIM CU’O’NG TRONG BAT DANG THUC CAN DAI 223 §6. Bit ding thirc hoin vi vi K9 thuit su’ dung 225 §6.1. Gidi thiéu v'é bit ding thirc hoin vi Z25 §6.2. Ky thuit sir dung bit ding thirc hoin vi 229 §7. Bit ding thirc Schur vi ky thuit sir dung 249 §7.1. Gidi thiéu vé bit ding thu’c Schur 249 §7.2. Ky thuit sir dung bit ding thllrc Schur Z54 Ung dung bit ding thin: Schur trong chirng minh bit ding thirc §7.3. -.. . .1. 265 don xu’ng ba blen §s. Djnh ly Muirhead vi bit ding thirc d6i xu’ng 279 §s.1. Gié'i thiéu dlnh I9 Muirhead 279 §8.2. Ky thuit s\'r dgng dinh ly Muirhead" 288 CHU’O’NG Ill: NHCPNG VIEN KIM cuowe TRONG GIAI TlCH 307 §9. Dinh ly Fermat vi (mg dung trong bit ding thvirc 309 §9.1. Gidi thléu dinh 19 Fermat 1 ' 309 §9.2. U'ng dung dlnh ly Fermat 311 §10. Djnh ly Lagrange vi cic (rng dung trong bit ding thirc 339 §10.1. Dlnh ly Lagrange cho him mét bié'n vi cac u’ng dung 339 §10:2. Cu'c trl cua him nhi'éu biin vi phu’dng phip nhin tu’ Lagrange 347 §10.2.1. Cgrc trj khéng c6 dléu kién ring buéc 347 §10.2.2. Cu’c tri c6 dféu kién ring bu
4 §11.2. K? thuit chgn diim rdi trong bit ding thirc Bernoulli 374 §12. Bit ding thirc Jensen vi k7 thuit sir dung 393 §12.1. Him léi, him l6m vi bit ding thirc Jensen 393 §12.2. K? thuit sir dqng bit ding thirc Jensen 399 §13. Bit ding th1'rc Karamata vi k9 thuit sir dgng 411 §13.1. Gidi thiéu ve bit ding thfrc Karamata 411 §13.2. Ky thuit su' dgng bat dang thirc Karamata 418 §14. Bit ding thirc vé’i céc him s6 léi bén phii vi l6m bén tréi 425 §14.1. Céc djnh Iv v‘é him s6 l6i bén phii vi l6m bén trii 425 §14.2. KY thuit sir dgng bit ding thirc RCF, LCF, LCRCF 43O §15. Bit ding thirc Popoviciu 457 §16. Bit ding'thL'rc trong tlch phin Riman 465 CHU’O’NG IV: NHUNG VIEN KlM cuowe TRONG BAT DANG THl}‘C HIEN DAI 513 §17. Phu'dng phép phin tich ting binh phlfdI1g(SOS) 515 §18. Phlrdng phép d6n bié'n (MV) 549 §19. Phu’o’ng phép ABC 627 §20. Phtrdng phép hinh hqc héa dii s6 (GLA) 681 §21. Phu’dng phép EV 735 §22. Phu'dng phép chia dé trj (DAC) 771 CHLPO’NG v: MQT so SANG TAO vE BAT DANG THU’C 805 §23. NhCI'ng bii vié't chqn lqc vé bit ding thllrc 805 §23.1 vé mr diy bit ding thirc bic ba vi (mg dgng 805 §23.2 D6i bié'n s6 di sing tio vi chfrng minh bit ding thirc 809 §23.3 Phu’0’ng phép dinh gié him $6 tii bién 817 §23.4 Phnrdng phép tié'p tuyin chirng minh bit ding thirc 829 §23.s Phu'0‘ng phép hé $6 bit dinh 833 §23.6 Céc phép d6i bié'n thuin nghich theo cic dé dii trong tam giic 879 §23.7 Phu‘dng phép dinh gii céc hé s6' cda da thc bing djnh Iv Viéte 891 §23.2; Bit ding rm'r¢ kh6ng thuin nhit 905 §23.9 Phu’dng phip SS 915 §23.1O Céc ting (I61 x|.'1’ng vi bit ding thirc hoén vi 933 §24. NhG’ng bit ding thirc chgn Iqc 949 §24.1 ca¢ bii toin cé nhléu lei giii 949 §24.2 Vé mét bit ding thirc thi toin quic té, 977 §24.3 Ciu chuyén v‘é bit ding thc Nesbitt - Shapiro 987 §24.4 Bit ding thirc Jack Garfunkel vi mét s6' m6 ring 1015 §24.5 Cic bit ding thrc cé ldi giii hay 1027 CHU’O’NG VI: TONG KET 1055 §25.1 Tém tit nhfrng vién kim cnrdng vi cic bit ding thfrc co’ bin 1055 §25.2 Céc bit ding thdc chqn lgc dinh cho bin dgc 1089 §25.3 Nhin lii vi m6 ra 1099 PHU LUC: Th6ng ké bii viét, bii toin sir dung vi tii liéu tham khio 1114
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