SHARP INEQUALITIES INVOLVING THE CONSTANT e AND THE SEQUENCE + / NECDET BATIR Abstract. Several ew ad sharp iequalities ivolvig the costat e ad the sequece + / are proved.. INTRODUCTION The costat e or Euler s umber which is usually represeted by e = lim + or e = =0! is oe of the most importat costats i mathematics ad sciece. I additio to may other applicatios, it is ivolved i some mathematical iequalities. For example, the well-kow Carlema s iequality[3] a a 2...a / < e a, = = where a 0 =, 2,... ad = a <, ad its Polya s geeralizatio[2] /σ λ a λ aλ2 2...aλ < e λ a, = = where λ > 0, σ = k= λ k, a 0 =, 2,... ad 0 < = λ a <, are ice examples of applicatios of approximatio of e. Recetly, some authors have obtaied iterestig iequalities for this importat costat ad applied them to refie Hardy ad Carlema s iequalities. We list here some of them: Zitia, ad Yibig[9] Sador[9] e 7 < + x < e 4x + 2 x e + x /2 exp 32x + 2 < + x < e 6 2x + x 2x + 2x + 2 exp x, 62x + 2 2000 Mathematics Subject Classificatio. Primary: 26A09; Secodary: 33B0; 26D99. Key words ad phrases. Costat e, Euler umber, logarithmic fuctio ad iequalities.
2 NECDET BATIR ad Yag[5, Lemma] + x x [ ] < e 2 + x 24 + x 2 48 + x 3. Also, i [0] ad [] Sador proved the followig ice iequalities: + +α < e < + +β, where α = = 0.44269...ad β = 0.5 are best possible costats, ad + a < e + / < + b,. with best possible costats a = e/2 ad b = /2, respectively. Some other similar results ca be foud i [4, 5, 2, 4, 6, 7, 8]. I this ote we aim to establish some more ew ad sharp iequalities for this importat costat ad the sequece + /. Sice some iequalities we used i the proofs of our mai results are simple cosequeces of iequalities for some special meas, we wat to recall them here briefly. The arithmetic, geometric, logarithmic ad idetric meas of the positive real umbers a ad b are defied as: A = Aa, b = a + b/2, G = Ga, b = ab, L = La, b = b a/log b log a ad I = Ia, b = /eb b /a a /b a, respectively. These meas satisfy the followig iequalities: G L,.2 which is due to Carlso[], ad L I,.3 which is due to Stolarsky, see[3]. For may other iterestig iequalities for these importat meas refer to [6, 7, 8]. 2. MAIN RESULTS Now we are i a positio to establish our mai results. Theorem 2.. For all positive itegers, we have e + log + / < + 4 + log + /, 2. where the costats e = 2.7828... ad 4log2 = 2.77259... are best possible. Proof. Defie for x > 0 ηx = x + x+ x x log + /x. 2.2
3 Differetiatio yields η x = x + x+ x x + x x 2. + x Sice ad Lx, x + = log + /x, Gx, x + = x 2 + x, by.2 we have + /x < x 2 + x. 2.3 So, we fid that η x < 0 for x > 0. We ca easily show that lim x ηx = e, cocludig for =, 2, 3,... e = η < η η = 4. 2.4 Replacig the value of η here proves Theorem 2.. Sice e.ix, x + = x + + /x x, where I is idetric mea, we wat to ote that the left had side of 2. ca be derived by usig the mea iequality Lx, x + Ix, x + give i.3. Theorem 2.2. For all positive itegers the followig iequalities hold: 4 + log + / where αx = x 2 + x + /x. α < + 4 + log + / Proof. By mea value theorem we have a φ such that 0 < φ = φt < ad Defie log + /t = ρt = α,, t > 0. 2.5 t + φt e /t. 2.6 It is ot difficult to verify that φ ρt = t ρt. 2.7 Itegratig both sides of 2.5 over t x, we get
4 NECDET BATIR x dt = x + log x + x log x 2. 2.8 t + φt Iducig the chage of variable t = ρu here, where ρ is as defied by 2.6, ad usig 2.7 we get for x x + φx / ρ u du u Differetiatio of 2.6 twice yields = x + log x + x log x 2. 2.9 e u 3 ρ /u = u 3 e u ue u 2e u + u + 2 = u 3 e u k=3 k 2 u k > 0. k! Hece, ρ is strictly icreasig o 0,. Sice ρ u is positive for u > 0, we obtai from 2.9 that ρ / log + log x + φx < x+ logx+ x log x 2 < ρ x + φ x log + log x + φ x. Sice x + φx = log + /x, simplifyig these iequalities we get for x log + /x provig Theorem 2.2. 2 2 < x + 4 + x x < log + /x x 2 +x +/x, Corollary 2.3. For all positive itegers the followig double iequality holds: 4 + log + / a + 4 + log + / where a = 2 2= 0.960906... ad b = are best possible costats. Proof. From 2.0 we get for x 2 2 log log + /x By 2.3 we get for x x + x+ < log 4x x < x 2 + x + /x log 2.0 b, 2. log + /x.
5 2 2 log log + /x x + x+ < log 4x x < log log + /x This proves that 2. is satisfied for a = 2 2 ad b =. Now we assume that the right iequality of 2. holds. The we have to have b lim log + 4 + =. log log+/ Similarly, from the left iequality of 2. we ca write log + 4 + log log +/ a lim = 2 2, so that the costats a ad b are best possible, completig the proof of the Corollary. I the followig we establish a compaio of.. Theorem 2.4. For all positive itegers the followig iequalities hold: e + a < + + < e + b, 2.2. where a = 4/e = 0.475776... ad b = 0.5 are best possible costats. Proof. Let ϕx = x + x+ x x e x, x > 0. 2.3 Differetiate 2.3 to get ϕ x = e [ x + x+ x x log + /x e ] = e ηx e, where η is as defied i 2.2. By 2.4 we have ηx > e for x. By this fact, we arrive at that ϕ is strictly icreasig o 0,. Oe ca easily show that lim x ϕx = /2. Hece we have for =, 2, 3,... 4/e = ϕ < ϕ < ϕ = /2, from which the proof follows. Theorem 2.5. Let be a iteger. The we have exp 2 + c 2 < + < exp 2 + d 2, 2.4
6 NECDET BATIR where the costats c = = 0.27649... ad d = /3 = 0.33333... 2 2 are best possible. Proof. Defie for x > 0 ωx = 2 x log x. 2.5 + x We shall show that ω is strictly icreasig o 0,. I order to fulfill this it is eough to show that ω /x > 0 for x > 0. Now we have x 2 ω /x = 2x logx + 3/2 x 3 /x + x 2 2x 2 log x + 3/2. Hece, i order to show ω /x > 0 we oly eed to show or equivaletly 2x 2 logx + 3/2 x3 x + < 0, vx := 2x + 2/3 x logx + x 2 < 0. Differetiatio successively we get v0 = v 0 = v 0 = 0 ad v x = 4 27 x + 7/3 5x + 4 logx + < 0, which proves vx < 0 for x > 0. So, ω is strictly icreasig o 0,. Oe ca easily check that lim x ωx = /3, cocludig for =, 2,3,... c = 2 2 = 0.27649... = ω ω < ω = /3 = d. Usig 2.5 ad the simplifyig this iequality we prove Theorem 2.5. Theorem 2.6. Let be a iteger. The the followig double iequality holds where exp + < + + < exp + 2 + α α = /3 = 0.3333333... ad β = are best possible costats., 2.6 2 + β = 0.2943497... 2.7 4 2
7 Proof. We make the followig auxiliary fuctio for x ad a > 0 gx, a = x + log + x 2x + a. 2.8 By differetiatio with respect to x successively, we fid that for x ad a > 0 ad g x, a = log + x x + 2x + a 2 2.9 g x, a = 3a x2 + 3xa 2 + a 3 x 3 + x 2 x + a 3. 2.20 Now from 2.20 We get g x, α = 3 x + 27 x 3 + x 2 x + > 0, 3 3 where α is as give i 2.7. Hece, x g x, α is strictly icreasig o [,. But sice lim x g x, α = 0, we have g x, α < 0 for x. This implies that x gx, α is strictly decreasig o [,. From the fact that lim x gx, α = 0, we obtai gx, α > 0 for x, provig the left iequality of 2.6. From 2.20 we fid that g x, β = a 0x 2 + a x + a 2 x 3 + x 2 x + β 3, 2.2 where a 0 = 0.69509..., a = 0.25992524... ad a 2 = 0.02502972... It is easy to see from 2.2 that for x 3 g x, β < 0, that is, x g x, β is strictly decreasig o [3,. Sice lim x g x, β = 0, this yields g x, β > 0 for x 3. But this meas x gx, β is strictly icreasig o [3,. Sice lim x gx, β = 0, we get gx, β < 0 for x 3. A simple calculatio gives g, β < 0 ad g2, β < 0. Therefore, for ay positive iteger we have g, β < 0. This proves the right iequality of 2.6. From the right of 2.6 we get β lim 2 [ + log + ]. It is easy to evaluate this limit ad to show that has value /3, hece we have β /3. Similarly, from the left iequality of of 2.6 we get for all positive itegers which yields α lim 2 [ + log ] +, α 4 2 = 0.2943497... This proves that the costats α ad β give i 2.7 are best possible.
8 NECDET BATIR Theorem 2.7. Let be a iteger. The the followig double iequality holds where exp 2 + α = /4 = 0.25 ad β = 3 + α 3 < + < exp 2 + 3 + β 3, 2.22 = 0.9949... 2.23 3 3/2 3 are best possible costats. Proof. We defie for x ad t > 0 φx, t = log + /x x + 2x 2 3x + t 3. 2.24 Differetiatio of 2.24 with respect to x gives φ x, t = 4tx3 6t 2 x 2 4t 3 x t 4 x 3 x + x + t 4. 2.25 Hece, we fid that φ 24x 2 + 4x + x, /4 = 64x + 2 x + /4 4 < 0, so that x φx, /4 is strictly decreasig o,. Sice lim φx, /4 = 0, x this leads to φx, /4 > 0, ad the proof of the left of 2.22 follows from a simple calculatio. Similarly, from 2.25 we obtai that φ x, β = c 0 + c x 2 + c 2 x 2 2 + c 3 x 2 3 x 3 x + x + β 4, 2.26 where c 0 = 0.59609..., c =.46934..., c 2 = 0.97345..., ad c 3 = 0.20203..., ad β is as give i 2.23. Thus, we coclude that x φx, β is strictly icreasig for x 2. But sice φx, β = 0, we get φx, β < 0 for x 2. A easy lim x computatio gives φ, β = 0, so that we get for all positive itegers, φ, β 0. This fiishes the proof of the right-had iequality i 2.22 by the help of 2.24. Now from the right-had iequality i 2.22 givig /3 /3 β lim, log + / / + /2 β. 3 3/2 3
9 By the same way we get from the left had iequality of 2.22 that /3 /3 α lim. log + / / + /2 It is ot difficult to prove that this limit goes to /4 as x goes to. These prove that the costats α ad β give i 2.23 are best possible. Ackowledgmets. I would like to thak the referee for useful suggestios ad his carefully readig the mauscript. Refereces [] B. C. CARLSON, Some iequalities for hypergeometric fuctios, Proc. Amer. Math. Soc.,7966,32-39. [2] G. H. HARDY, Notes o some poits i the itegral calculus, Messeger Math. 54, 50-56, 924-925. [3] M. JOHANSSON, L.E. PERSSON, ad A. WEDESTING, Carlema s Iequality-History, Proofs ad Some New Geeralizatios J. Iequal. Pure Appl. Math.JIPAM 43, Art.53, 2003. [4] S. KAIJSER, L-E. PERSSSON, ad A. OBERG, O Carlema ad Kopp s iequalities, J. Approx. Theory, 72002, 405. [5] J. L. LI, Notes o a iequality ivolvig the costat e, J. Math. Aal. Appl. 250 2000, 722-725. [6] E. NEUMAN, ad J. SANDOR, O certai meas of two argumets ad their extesios, It. J. Math. Math. Sci., 62003, 98-993. [7] E. NEUMAN, ad J. SANDOR, Iequalities ivolvig Stolarsky ad Gii meas, Math.Pao., 42003,o.,29-44. [8] J. SANDOR, O the idetric ad logarithmic meas, Aequatioes Mathematicae, 40990, 26-270. [9] J. SANDOR, Some itegral iequalities, Elemete der Math. Basel, 43988, 77-80. [0] J. SANDOR, O certai bouds for the sequece + / +a, Octogo Math. Mag., 32005,o., 906-907. [] J. SANDOR, O certai bouds for the umber e II, Octogo Math. Mag., 2003,o., 24-243. [2] J. SANDOR, ad L. DEBNATH, O certai iequalities ivolvig the costat e ad their applicatios, J. Math. Aal. Appl., 2492000, 569-582. [3] K. B. STOLARSKY, Geeralizatios of the logarithmic mea, Math. Mag., 48975, 87-92. [4] Y. XIAOJING, Approximatig for the costat e ad their applicatios, J. Math. Aal. Appl., 262200, 65-659. [5] Y. XIAOJING, O Carlema s iequality, J. Math. Aal. Appl., 253200,69-694. [6] P. YAN ad G. SUN,A stregtheed Carlema s iequality, J. Math. Aal. Appl. 240 999, 290-293. [7] B. YANG, O Hardy s iequality, J. Math. Aal. Appl. 234 999, 77-722. [8] B.YANG ad L. DEBNATH, Some iequalities ivolvig the costat e, ad a applicatio to Carlema s iequality, J. Math. Aal. Appl. 223 998, 347-353. [9] X. ZITIAN, ad Z. YIBING, A best Approximatio for the costat e ad a improvemet to Hardy s iequality, J. Math. Aal. Appl., 2522000, 994-998. Departmet of Mathematics, Faculty of Arts ad Scieces, Yuzucu Yil Uiversity, 65080, Va, Turkey E-mail address: ecdet batir@hotmail.com