Filomat 29:7 205, 535 539 DOI 0.2298/FIL507535M Published by Faculty of Scieces Mathematics, Uiversity of Niš, Serbia Available at: http://www.pmf.i.ac.rs/filomat Estimates of + x /x Ivolved i Carlema s Iequality Keller s Limit Cristiel Mortici a, X.-J. Jag b a Valahia Uiversity of Târgovişte, Bd. Uirii 8, 30082, Târgovişte, Romaia Academy of Romaia Scietists, Splaiul Idepedeţei 54, 050094 Bucharest Romaia b Departmet of Mathematics, Tsighua Uiversity, Beijig 00084, Chia Abstract.The aim of this work is to exted the results obtaied by Yag Bicheg L. Debath i [Some iequalities ivolvig the costat e a applicatio to Carlema s iequality, J. Math. Aal. Appl., 223 998, 347-353]. We preset a simple proof of our ew result which ca be also used as a direct proof for Yag Bicheg L. Debath results. Fially some applicatios to geeralized Keller s it further directios are provided.. Itroductio Motivatio Yag Bicheg L. Debath [4, Lemma 2.] preseted the followig double iequality for every x i 0 < x 5 : e 2 x + e 24 x2 2e 48 x3 < + x /x < e 2 x + e 24 x2. Such iequalities were prove to be of great iterest through the researchers, especially i the recet past, due to may practical problems they ca be applied. As example, we refer to iequality which is the mai tool for improvig Carlema s iequality i [4]. As i [4] it is provided a log, difficult proof, we propose i this paper a simple, direct proof of. The proof we provide shows us that is true for every real umber x 0, ]. We show our method i case of the followig improvemet of. Theorem. For every real umber x 0, ], we have a x < + x /x < b x, 2 a x = e 2 x + e 24 x2 2e 48 x3 + 2447e 5760 x4 959e 2304 x5 200 Mathematics Subject Classificatio. Primary 26A09; Secodary 33B0, 26D99 Keywords. Costat e; Carlema s iequality; Keller s it; approximatios Received: 28 December 203; Accepted: 02 July 204 Commuicated by Hari M. Srivastava Ackowledgmets: The work of the first author was supported by a grat of the Romaia Natioal Authority for Scietific Research, CNCS-UEFISCDI project umber PN-II-ID-PCE-20-3-0087. Email addresses: cristiel.mortici@hotmail.com Cristiel Mortici, yagxj@mail.tsighua.edu.c X.-J. Jag
C. Mortici, X.-J. Jag / Filomat 29:7 205, 535 539 536 b x = a x + 959e 2304 x5. 2. The Proof of the Theorem Proof of Theorem. Iequalities 2 are equivalet to f < 0 > 0 o [,, f x = x l + l x 2x + 24x 2 2 48x + 2447 3 5760x 4 x = x l + l x 2x + 24x 2 2 48x + 2447 3 5760x 959. 4 2304x 5 We have f A x x = x 2 x + 2 P 2 x < 0 x = B x x 2 x + 2 Q 2 x > 0, A x = 2597 48 593x + 4933 395 36x 2 + 4673 488 800x 3 +2204 426 400x 4 + 44 288 000x 5 + 57 054 93 P x = 7 60x + 28 560x 2 + 20 60x 3 + 5760x 4 + 5447 B x = 7 596 034 39x + 42 04 67 375x 2 + 53 575 389 600x 3 +38 240 425 36x 4 + 4 508 374 400x 5 + 2285 22 800x 6 + 2934 528 883 Q x = 45 24x + 9 440x 2 + 97 440x 3 + 5 840x 4 + 520x 5 + 6099. As f is strictly cocave is strictly covex o, with f = = 0, we deduce that f < 0 > 0 o,. As f = l 2 l 5447 = 0.250 98... < 0 5760 = l 2 l 2033 = 0.329... > 0, 3840 the coclusio follows.
3. Keller s it Geeralizatio C. Mortici, X.-J. Jag / Filomat 29:7 205, 535 539 537 As a applicatio, a ew proof of the it + = e 3 also kow as Keller s it, see e.g. [3] ca be costructed. Ideed, usig Theorem, we get + a b + < < + b a. Extreme-side expressios are ratioal fuctios i havig e as commo it, so 3 is true. Moreover, 2 + a b 4 + < 2 < 2 + b a. With some patiet, or better usig a computer software such as Maple, we obtai 2 + a b 2 + b a = = U = 9 279 29 32 2 + 20 598 3 7507 4 77 5 920 6 + 480 7 4795, U 520 3 4 e 5 V 520 2 5 e 6 V = 24 66 49 90 2 + 52 080 3 24 970 4 +9793 5 2400 6 + 480 7 4894. As expressios i 5-6 teds to e/24, 4 reads as + 2 = e 24. 7 Usig the same method we discovered preset ow the followig ew results. Theorem 2. For every c R, we have + + +c +c + = e 8 + c + c
C. Mortici, X.-J. Jag / Filomat 29:7 205, 535 539 538 2 + + +c +c + = e 2c. 9 + c + c 24 I c = /2 case, we have 3 + + + 2 + 2 + 2 2 Limits 3 7 are case c = 0 i 8, respective 9. By aalogue remarks, we have + a b + c + c < + + +c +c + + c + c < + b a. + c + c + a + c = + b + c = e, b a + c + c = 5e 44. 0 as the ivolved sequeces are ratioal fuctios i. The exact forms were obtaied ca be verified by the reader usig a computer software such as Maple. Fially, 2 + a + c < 2 + + < 2 + b + c + c b + c +c + a + c +c + c the extreme-side sequeces coverge to e 24 2c. The sequece 0 is bouded below above by u = 3 + a + b 2, 2 respective v = 3 + b + a 2, 2 6e 3 P u = 5 2 + 5 2 4
C. Mortici, X.-J. Jag / Filomat 29:7 205, 535 539 539 v = 6e 3 Q 5 2 + 4 2 5, with P = 50 380 44 7958 707 488 2 + 528 087 04 3 280 800 52 4 2395 32 46 5 + 49 299 200 6 64 676 909 Q = 5884 739 3702 534 662 304 2 + 3 665 440 448 3 4838 973 696 4 + 035 06 704 5 + 49 299 200 6 088 865 8. Now easy, u = v = 5e/44. Fially, we are coviced that the iequalities preseted i Theorem ca be succesfully used to obtai other ew results, such as those preseted i this paper, but also i the problem of improvig iequalities of Carlema s type. See recet works [], [3]-[6]. Remark 3.. Computatios made i this paper were performed usig Maple software. The latest chages of this paper were completed durig the first author s visit to Uiversity of Jae, Spai. Refereces [] Ya Pig, Guozheg Su, A stregtheed Carlema s iequality, J. Math. Aal. Appl., 240 999, 290-293. [2] G. Pólya, G. Szegö, Problems Theorems i Aalysis, vol. I, Spriger Verlag, New York, 972. [3] J. Sor, O certai iequalities ivolvig the costat e their applicatios, J. Math. Aal. Appl., 249 2000, 569-582. [4] Bicheg Yag, L. Debath, Some iequalities ivolvig the costat e a applicatio to Carlema s iequality, J. Math. Aal. Appl., 223 998, 347-353. [5] Xiaojig Yag, O Carlema s iequality, J. Math. Aal. Appl., 253 200, 69-694. [6] Hu Yue, A stregtheed Carlema s iequality, Commu. Math. Aal., 2006, o. 2, 5-9.