Generalizations and analogues of the Nesbitt s inequality

Transcription

1 OCTOGON MATHEMATICAL MAGAZINE Vol 17, No1, Apil 2009, pp ISSN , ISBN , wwwhetfaluo/octogo 215 Geealiatios ad aalogues of the Nesbitt s iequalit Fuhua Wei ad Shahe Wu 19 ABSTRACT The Nesbitt s iequalit is geealied b itoducig epoet ad weight paametes Seveal Nesbitt-tpe iequalities fo vaiables ae povided Fiall, two aalogous foms of Nesbitt s iequalit ae give 1 INTRODUCTION The Nesbitt s iequalit states that if,, ae positive eal umbes, the 3 2, 1) the equalit occus if ad ol if the thee vaiables ae equal [1], see also [2]) It is well kow that this cclic sum iequalit has ma applicatios i the poof of factioal iequalities I this pape we shall establish some geealiatios ad aalogous foms of the Nesbitt s iequalit 2 GENERALIZATIONS OF THE NESBITT S INEQUALITY Theoem 1 Let,,, k be positive eal umbes The k k k 3 1 k 2) Poof B usig the Cauch-Schwa iequalit see [3]), we have k k k ) 2 k 2 k 2 k ) 2 ) 19 Received: Mathematics Subject Classificatio 26D15 Ke wods ad phases Nesbitt s iequalit; Cauch-Schwa iequalit; Chebshev s iequalit; powe mea iequalit; geealiatio; aalogue

2 216 Fuhua Wei ad Shahe Wu Hece k k k ) 2 1 k) ) The Theoem 1 is poved k) ) 3 1 k Theoem 2 Let 1, 2,, be positive eal umbes, 2 The Poof Let s 1 2, oe has ) s 1 s 2 s B smmet, we ma assume that 1 2, the s 1 s 2 s, Usig the Chebshev s iequalit see [3]) gives 1 1 s 1 2 s s 1 ) 2 s 2 ) s ) s 1 s 2 s 1 s 1 2 s 2 o equivaletl s 1 s 1 2 s 2 this is eactl the equied iequalit s ) [s 1 ) s 2 ) s )], s 1,

3 Geealiatios ad aalogues of the Nesbitt s iequalit 217 Theoem 3 Let 1, 2,, be positive eal umbes, 2, k 1 The ) Poof Usig the powe mea iequalit ad the iequalit 3), we have ) k 1 k i s i i1 1 This completes the poof Theoem 4 Let 1, 2,, be positive eal umbes, ad let λ 1, s > 0, p The s i i1 i i1 p s i 1 λ 1 p Poof Usig the powe mea iequalit see [3]), we have s 1) 5) i 1 λ i i1 p s i i1 p s i O the othe had, b smmet, we ma assume that 1 2, the s 1 s 2 s > 0, p s p s 1 p s 1 > 0 Applig the geealied Rado s iequalit see [4-7]) i1 a α i b i 2 α a i ) α / b i ) a 1 a 2 a > 0, b b 1 b 1 > 0, α 1), we deduce that i1 i1

4 218 Fuhua Wei ad Shahe Wu i1 Theefoe i p s i i1 s i ) s p s i 2 s s i ) s i1 p ) s 1, p s i ) 1 i1 i i1 p s i 1 λ i1 i p s i The poof of Theoem 4 is complete 1 λ 1 p s 1 ) I Theoem 4, choosig λ 1, s 1, 3, 1, 2, 3, we get Theoem 5 Let,, be positive eal umbes, ad let p, 1 The 3 p ) 1 6) 2 3 I paticula, whe 1, the iequalit 6) becomes the Nesbitt s iequalit 1) 3 ANALOGOUS FORMS OF THE NESBITT S INEQUALITY Theoem 6 Let,, be positive eal umbes, The Poof Note that ) ) ) ) ) ) B usig the Cauch-Schwa iequalit, we have ) ) ) ) 7)

5 Geealiatios ad aalogues of the Nesbitt s iequalit 219 [ ) ) ) ) 2 ) ) ) ) ) ) Thus, to pove the iequalit 7), it suffices to show that [ ] ) ) ) ) ) ) ) Diect computatio gives [ ) ) ) ) ) ) ) 9 4 ] 9 4 ] )[ ) ) )] 9 ) ) ) 4 4 )[ ) ) )] 9 ) ) ) 4 ) ) ) 8 ) ) 9 ) ) ) 4 ) ) ) , 4 ) ) ) whee the iequalit sig is due to the aithmetic-geometic meas iequalit The Theoem 6 is thus poved Theoem 7 Let,, be positive eal umbes, α 1/2, The ) α ) α ) α 3 2 α 8) Poof It follows fom the powe mea iequalit that ) α ) α ) α 3 1 2α ) 2α ) 3 2α 3 1 2α α

6 220 Fuhua Wei ad Shahe Wu The iequalit 8) is poved Remak The iequalit 8) is the epoetial geealiatio of iequalit 7) As a futhe geealiatio of iequalit 7), we put fowad the followig the followig cojectue Cojectue Let 1, 2,, be positive eal umbes, 2, α 1/2 The ) α ) α ) α ) α α 9) Ackowledgemets The peset ivestigatio was suppoted, i pat, b the iovative epeimet poject fo uivesit studets fom Fujia Povice Educatio Depatmet of Chia ude Gat No214, ad, i pat, b the iovative epeimet poject fo uivesit studets fom Loga Uivesit of Chia REFERENCES [1] Nesbitt, A M, Poblem 15114, Educatioal Times, ), [2] Dâmbe, M O, Iequalities - Ideas ad Methods, Ed Gil, Zalǎu, 2003 [3] Mitiović, D S ad Vasić, P M, Aaltic Iequalities, Spige-Velag, New Yok, 1970 [4] Wu, Sh-H, A epoetial geealiatio of a Rado iequalit, J Huaqiao Uiv Nat Sci Ed, 24 1) 2003), [5] Wu, Sh-H, A esult o etedig Rado s iequalit ad its applicatio, J Guihou Uiv Nat Sci Ed, 22 1) 2004), 1 4 [6] Wu, Sh-H, A ew geealiatio of the Rado iequalit, Math Pactice Theo, 35 9) 2005), [7] Wu, Sh-H, A class of ew Rado tpe iequalities ad thei applicatios, Math Pactice Theo, 36 3) 2006), Depatmet of Mathematics ad Compute Sciece, Loga Uivesit, Loga, Fujia , pr Chia wushahe@ahoocomc