Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming Chu, Miao-Kun Wang, and Ye-Fang Qiu Depatment of Mathematics, Huzhou Teaches College, Huzhou 313000, China Coespondence should be addessed to Yu-Ming Chu, chuyuming005@yahoo.com.cn Received 3 May 011; Accepted 16 August 011 Academic Edito: Dik Aeyels Copyight q 011 Yu-Ming Chu et al. This is an open access aticle distibuted unde the Ceative Commons Attibution License, which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited. We pove that the double inequality / ath/ 3/4 α < K < / ath/ 3/4 β holds fo all 0, 1 with the best possible constants α 0andβ 1/4, which answe to an open poblem poposed by Alze and Qiu. Hee, K is the complete elliptic integals of the fist kind, and ath is the invese hypebolic tangent function. 1. Intoduction Fo 0, 1, Lengede s complete elliptic integals of the fist and second kind 1 ae defined by K K / 0 1 sin θ 1/ dθ, K K K ( ), K 0, K 1, E E / 0 1 sin θ 1/ dθ, E E E ( ), 1.1 E 0, E 1 1,
Abstact and Applied Analysis espectively. Hee and in what follows, we set 1. These integals ae special cases of Guassian hypegeometic function F a, b; c; x F 1 a, b; c; x n 0 a, n b, n x n c, n n! 1 <x<1, 1. whee a, n n 1 k 0 a k. Indeed, we have K F ( 1, 1 ;1; ), E F ( 1, 1 ;1; ). 1.3 It is well known that the complete elliptic integals have many impotant applications in physics, engineeing, geometic function theoy, quasiconfomal analysis, theoy of mean values, numbe theoy, and othe elated fields 13. Recently, the complete elliptic integals have been the subject of intensive eseach. In paticula, many emakable popeties and inequalities can be found in the liteatue 3, 10 18. In 199, Andeson et al. 15 discoveed that K can be appoximated by the invese hypebolic tangent function, ath, and poved that ) 1/ < K < fo 0, 1. In 16, Alze and Qiu poved that the double inequality ) α < K < ), 1.4 ( ) ath β, 1.5 holds fo all 0, 1 with the best possible constants α 3/4 andβ 1 and poposed an open poblem as follows. Open Poblem # The double inequality ) 3/4 α < K < ) 3/4 β, 1.6 holds fo all 0, 1 with the best possible constants α 0andβ 1/4. It is the aim of this pape to give a positive answe to the open poblem #.. Lemmas and Theoem In ode to establish ou main esult, we need seveal fomulas and lemmas, which we pesent in this section.
Abstact and Applied Analysis 3 Fo 0 <<1, the following deivative fomulas wee pesented in 4, Appendix E, pages 474-475 : dk d E K, ( ) d E K K, d de d E K, d K E d E..1 Lemma.1 see 4, Theoem 1.5. Fo <a<b<,letf, g : a, b R be continuous on a, b and be diffeentiable on a, b,letg x / 0 be on a, b.iff x /g x is inceasing (deceasing) on a, b,thensoae f x f a g x g a, f x f b g x g b.. If f x /g x is stictly monotone, then the monotonicity in the conclusion is also stict. The following Lemma. can be found in 9, Lemma 3 1 and 4, Theoem 3.1 1 and Execise 3.43 30 and 46. Lemma.. 1 c ath / is stictly deceasing in 0, 1 if and only if c /3; E K / is stictly inceasing fom 0, 1 onto /4, 1 ; 3 E K / K is stictly deceasing fom 0, 1 onto 0, 1/ ; 4 K/ ath is stictly deceasing fom 0, 1 onto 1,/. Lemma.3. 1 f 1 ath / 3 is stictly inceasing fom 0, 1 onto /3, 1 ; f log ath / / is stictly inceasing fom 0, 1 onto 1/3, ; 3 f 3 E ath K ath /4 3K/4 / 5 is stictly inceasing fom 0, 1 onto /480, ; 4 f 4 3/4 /4 ath K E K ath is positive and stictly inceasing in /, 1 ; 5 f 5 3/4 log ath / log K/ is positive and stictly inceasing on 0, 1/4. Poof. Fo pat 1, leth 1 ath and h 3. Then f 1 h 1 /h, h 1 0 h 0 0and h 1 h 3 ath..3 It is well known that the function ath / is stictly inceasing fom 0, 1 onto 1,. Theefoe, fom.3 and Lemma.1 togethe with l Hôpital s ule, we know that f 1 is stictly inceasing in 0, 1, f 1 0 /3 andf 1 1 1.
4 Abstact and Applied Analysis Fo pat, clealy f 1. Leth 3 log ath / and h 4, then f h 3 /h 4, h 3 0 h 4 0 0, and h 3 ath ath 1 h 4 ath 3 ath..4 It follows fom Lemma.1, Lemma. 1,pat 1,.4,andl Hôpital s ule that f is stictly inceasing in 0, 1 and f 0 1/3. Fo pat 3,fomLemma. 4, we clealy see that f 3 1.Leth 5 E ath K ath /4 3K /4, h 6 5,h 7 E K / 4 K ath / 3ath E K / 4,andh 8 4, then f 3 h 5 /h 6, h 5 0 h 6 0 h 7 0 h 8 0 0, h 7 1 h 8 4 4 h 5 h 6 1 h 7 5 h 8, ath 3 [ 3 E K 1 4 4 E ]..5 Fom Lemma. and pat 1, we clealy see that h 7 /h 8 is stictly inceasing in 0, 1. Thus, the monotonicity of f 3 can be obtained fom.5 and Lemma.1. Moeove, making use of l Hôpital s ule, we have f 3 0 /480. Fo pat 4, leth 9 1 E/ K 3 E K / K. Then, Lemma. 3 leads to the conclusion that h 9 is stictly inceasing in 0, 1. Notethat f K E K 4 4 1 ( ) h 9 1.013 > 0, ( ) f 4 0.084 > 0, K ath h 9 > 4.6.7 ( ) K ath h 9 > 0.8 4 fo /, 1. Theefoe, pat 4 follows fom.7 and.8. Fo pat 5, simple computations lead to f 5 log lim 0 f 5 0, ) ( 3 4 ) ath ath E K K..9.10
Abstact and Applied Analysis 5 Making use of pats 1 4, one has K ath f K ath 4 5 f Kf 1 f 3 >Kf 1 f 3 > 3 f 3 ( ) 1 1.040 > 0 4.11 fo 0, 1/4. Theefoe, pat 5 follows fom.9 and.11. Lemma.4. Let ( ) [ ] ( ) 3 ath K g c c log log 4 c R,.1 then the following statements ae tue: 1 g c > 0 fo all 0, 1 if and only if c 1/4, ; g c < 0 fo all 0, 1 if and only if c, 0. Poof. Fistly, we pove that g c > 0foc 1/4,. Since g c is continuous and stictly inceasing with espect to c R fo fixed 0, 1, itsuffices to pove that g 1/4 > 0 fo all 0, 1. Notethat g 1/4 1 4 log lim 0 g 1/4 0, ) ( 3 4 1 ) 4 ath E K ath K..13.14 We divide the poof into two cases. Case 1 0, /. Then, making use of Lemma.3 1 3 and.14, we have K ath g K ath 3 1/4 4 f 1 4 K f 1 f 3 > 1 4 K f 1 f 3 > 1 f 3 ( ).15 0.50 > 0. Case /, 1. Then, making use of Lemma.3 4 and.14, weget g 1/4 log ath / 1 4 f 4 K ath log ath / > 0..16 Inequalities.15 and.16 imply that g 1/4 is stictly inceasing in 0, 1. Theefoe, g 1/4 > 0 follows fom.13 and the monotonicity of g 1/4.
6 Abstact and Applied Analysis On the othe hand, inequality 1.5 leads to the conclusion that g c < 0 fo all 0, 1 and c, 0. Next, we pove that the paametes 1/4 and 0 ae the best possible paametes in Lemma.4 1 and, espectively. If c 0, 1/4, then g c c f 5 c > 0 follows fom Lemma.3 5. Moeove, let F g c log ath / 3 c log K/ 4 log ath /,.17 then, using l Hôpital s ule and Lemma. 4, weget lim 1 F c 1 4 < 0..18 Inequality.18 implies that thee exists δ δ c > 0 such that F < 0 fo all 1 δ, 1. Theefoe, g c < 0fo 1 δ, 1 follows fom.17. Fom Lemma.4, we clealy see that the following Theoem.5 holds, which give a positive answe to the open poblem #. Theoem.5. The double inequality ) 3/4 α < K < ) 3/4 β.19 holds fo all 0, 1 with the best possible constants α 0 and β 1/4. Acknowledgments This pape was suppoted by the Natual Science Foundation of China unde Gant 11071069 and the Innovation Team Foundation of the Depatment of Education of Zhejiang Povince unde Gant T0094. Refeences 1 F. Bowman, Intoduction to Elliptic Functions with Applications, Dove Publications, New Yok, NY, USA, 1961. P. F. Byd and M. D. Fiedman, Handbook of Elliptic Integals fo Enginees and Scientists,Spinge-Velag, New Yok, NY, USA, 1971. 3 G. D. Andeson, S.-L. Qiu, and M. K. Vamanamuthy, Elliptic integal inequalities, with applications, Constuctive Appoximation, vol. 14, no., pp. 195 07, 1998. 4 G. D. Andeson, M. K. Vamanamuthy, and M. Vuoinen, Confomal Invaiants, Inequalities, and Quasiconfomal Maps, John Wiley & Sons, New Yok, NY, USA, 1997. 5 G. D. Andeson, M. K. Vamanamuthy, and M. Vuoinen, Distotion functions fo plane quasiconfomal mappings, Isael Jounal of Mathematics, vol. 6, no. 1, pp. 1 16, 1988. 6 M. Vuoinen, Singula values, Ramanujan modula equations, and Landen tansfomations, Studia Mathematica, vol. 11, no. 3, pp. 1 30, 1996. 7 G. D. Andeson, S.-L. Qiu, M. K. Vamanamuthy, and M. Vuoinen, Genealized elliptic integals and modula equations, Pacific Jounal of Mathematics, vol. 19, no. 1, pp. 1 37, 000.
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