Estimates of (1 + x) 1/x Involved in Carleman s Inequality and Keller s Limit

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1 Filomat 29:7 205, DOI /FIL507535M Published by Faculty of Scieces Mathematics, Uiversity of Niš, Serbia Available at: 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, 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., , ]. 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 x e 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 addresses: cristiel.mortici@hotmail.com Cristiel Mortici, yagxj@mail.tsighua.edu.c X.-J. Jag

2 C. Mortici, X.-J. Jag / Filomat 29:7 205, 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 x x 4 x = x l + l x 2x + 24x x x x 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 = x x x x x P x = 7 60x x x x B x = x x x x x x Q x = 45 24x x x x x 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 = < = l 2 l 2033 = > 0, 3840 the coclusio follows.

3 3. Keller s it Geeralizatio C. Mortici, X.-J. Jag / Filomat 29:7 205, 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 = , U e 5 V e 6 V = As expressios i 5-6 teds to e/24, 4 reads as + 2 = e 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

4 C. Mortici, X.-J. Jag / Filomat 29:7 205, c +c + = e 2c. 9 + c + c 24 I c = /2 case, we have 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 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 + 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 C. Mortici, X.-J. Jag / Filomat 29:7 205, v = 6e 3 Q , with P = Q = 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., , [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., , [4] Bicheg Yag, L. Debath, Some iequalities ivolvig the costat e a applicatio to Carlema s iequality, J. Math. Aal. Appl., , [5] Xiaojig Yag, O Carlema s iequality, J. Math. Aal. Appl., , [6] Hu Yue, A stregtheed Carlema s iequality, Commu. Math. Aal., 2006, o. 2, 5-9.