Geeral Mathematics Vol. 5, No. 2007), 75 80 About the use of a result of Professor Alexadru Lupaş to obtai some properties i the theory of the umber e Adrei Verescu Dedicated to Professor Alexadru Lupaş o his 65th aiversary Abstract A very elegat result of Professor Alexadru Lupaş gives us that the poit c of the mea value theorem of Lagrage) applied to the logarithmic fuctio o a iterval [a, b] 0, ) has the property that ab < c < a + b)/2. I this paper we show that the result of Professor Alexadru Lupaş is decisively useful to establish some other ice results i the theory of the umber e ad i some related questio. 2000 Mathematics Subject Classificatio: 26D07, 26D5. Key words ad phrases: The arithmetic, geometric ad harmoic meas, the umber e, the logarithm.. The work of Professor Alexadru Lupaş cotais umerous, various ad elegat results especially i the followig domais: Real fuctios, Iequalities, Sequeces ad Series, Special fuctios, Numerical Aalysis, Umbral Calculus, Approximatio Theory, but also Computer Scieces, Computatioal Mathematics ad Experimetal Mathematics. Received February, 2007 Accepted for publicatio i revised form) 3 March, 2007 75
76 Adrei Verescu All these result ca be foud i his umerous papers ad books. I this paper we cosider oe of his most elegat results, amely cocerig the positio of the poit,,c of the mea value theorem of Lagrage) applied to the logarithmic fuctio o a iterval [a, b] 0, ). This poit, give by the equatio: ) fb) fa) = f c) b a), for fx) = l x, x [a, b] 0, ) ad uique because of the ijectivity of the derivative has the property: 2) ab < c < a + b 2. This property refies the relatio c a,b), because, of course ) ab, a + b)/2 a,b). So it gives a aswer from a celebrated questio proposed by D. Pompeiu: to fid for the poit,,c smaller itervals as a,b) see [8], at the pages 20, 273, 293; also, see []). 2. To prove the two-sided iequality 2) cosider the followig Lemma [7], page 273). For ay x > 0, we have: 3) 2 2x + < l + ) < x xx + ). Proof. For the left part, cosider the fuctio f : 0, ) R, fx) = l + x). Because of the equality 2x+ f x) = xx )2x + ) < 0, 2 the fuctio f is strictly decreasig. The compariso with lim x fx) = 0, closes the proof.
About the use of a result of Prof. Alexadru Lupaş to obtai... 77 Aalogously, for the right part, cosider the fuctio g : 0, ) R, gx) = l + x). We obtai xx+) g x) = 2 2xx + ) xx + ) 2x + 2 ) xx + ) > 0 ad so, because of the compariso with lim gx) = 0, we obtai the right x part. Theorem. For ay iterval a,b) 0, ), a < b, cosiderig the poit,,c previously defied i the sectio, we have the iequality 2). Proof. The formula ), writte for the logarithmic fuctio becomes l b l a = b a ad 2) is equivalet with: c 4) 2b a) b + a < l b a < b a ab, or with: 5) 2t ) t + < l t < t t, where t = b/a ad t >. If we put e = + /s, with s > 0, the iequality 5) becomes e 2s + < l + ) <, s ss + ) which is exactly 3), writte i the variable s i the place of x, therefore it is true. We remark ow that, coversely, 3) implies 4). Ideed, puttig i 3) x = b, it gives us 4). So we have obtaied the a Theorem 2. The followig affirmatio are equivalet: a) The poit,,c of the mea theorem for the logarithmic fuctio, applied o a iterval [a,b] 0, ) satisfies the iequality 2). b) The iequality 3) is true. 3. The iequality 4) ca be also iterpreted i two maers for 0 < a < b):
78 Adrei Verescu a) It is a reffiemet of the classical iequality of Napier: b a b < l b a < b a a. b) The logaritmical mea of a ad b, amely b a l b/a the geometric ad the arithmetic meas of a ad b. is placed betwee 4. The property of the poit,,c of the logarithmic fuctio, give by Professor Alexadru Lupaş permits us to obtai other ice results. a) We have the iequality from [9]): 6) e 2 + 2 < e + ) < e 2 +. For a short proof, based o 3), see [3]. b)if γ = + + +... + l ad γ = lim γ 2 3 m = 0, 577... is the Euler, s costat, we have: 7) 2 + < γ γ < 2. A proof based o 3) ca be give i [2]. c) It is kow that the uique ratioal solutios of the equatio: are give by: x = e = x y = y x 0 < x < y) + ), y = f = + ) + =, 2, 3...) see [5], pp. 242-256). This meas that replacig these solutios i the equatio, we obtai a true equality, amely: )+) + + e f = f e =. Let z = )+) + + be. The sequece is strictly decreasig. For the proof, see [4]. This proof uses agai 3).
About the use of a result of Prof. Alexadru Lupaş to obtai... 79 A similar situatio is valid for the ratioal solutios of the equatio u u = v v 0 < v < u). We have u = e, v = f ad so: e ) ) )+) + e f = =. f + )+) + Let w = be. The sequece is also strictly decreasig; + the proof of [4] uses, agai, 3). d) The problem [6] of Professor Alexadru Lupaş is the followig:,,let x ad y be two differet real umbers x > 0, y > 0, x e so that x y = y x. Show that x y > e e. The solutio, published i the same joural i 2006, o. 3, pp. 233-236, also uses 3). e) I the paper [2], the sequeces of geeral term η = e le ) = = E le ) ad λ = le ) le ), where E = f = + ) +, are studied. I the proof of a part of its properties, 3) is also used. f) I the problem [3] the sequece x ) is defied by the equality + ) +x = e. The mootoy ad the covergece are requested. To solve the problem the iequality 3) are agai ecesary. A similar situatio is related to [0], where the properties of the sequece x ) give by the equality + 2 + 3 +...+ lx+x ) = γ are requested. The iequality 3) are agai used. Also se []. Refereces [] G. Şt. Adoie, The History of the Mathematics i Romaia i Romaia), Ed. of Scieces, Bucharest, 965. [2] E. Costatiescu, A. Lupaş ad A. Verescu, About two Sequeces related to Napier, s Costat, JIAT, Vol. 2007), No. to appear). [3] L. Lazarovici, B. Ramaza, The Problem 20383 i Romaia), Gazeta Mat. vol 00 985), No. 3, p. 94.
80 Adrei Verescu [4] A. Lupaş, The Problem 2739 i Romaia), Gazeta Mat. vol 78 973), No. 2, p. 08. [5] A. Lupaş, About the Theorem of the fiite icreasigs i Romaia), Rev. Mat. Timişoara, vol. 4 983), No. 2, 6-3. [6] A. Lupaş, The problem 203 i Romaia), Gazeta Mat. Series A), vol. 23 02), 2005, o., p. 280. 9 986). [7] D.S. Mitriović ad P.M. Vasić, Aalytic Iequalities, Spriger-Verlag, Berli Heidelberg New York, 970. [8] D. Pompeiu, The Mathematical Work i Romaia), Pritig House of the Romaia Academy, Bucharest, 957. [9] G. Pólya, G. Szegö, Problems ad Theorems i Aalysis, Spriger- Verlag, Berli Heidelberg New York, 970. [0] B. Ramaza, L. Lazarovici, The Problem C608, Gazeta Mat. vol 9 986). [] L.Tóth, About the Problem C608 i Romaia), Gazeta Mat. vol 94 989), 277-279. [2] A. Verescu, The Order of Covergece of the Sequece of Defiitio of the Number e i Romaia), Gazeta Mat. Series A), vol. 88 983), 580-28. [3] A. Verescu, A simple Proof of a Iequality related to the Number e i Romaia), Gazeta Mat. vol. 93 988), 205-206. [4] A. Verescu, O The Mootoy of some Sequeces cocerig the Equatios x y = y x ad u u = v v, Lucr. Sem. Didactica Matematicii, vol. 3 998), 229-232. [5] A. Verescu, The umber e ad the Mathematics of the Expoetial, Publishig House of the Uiversity of Bucharest, 2004 i Romaia).
About the use of a result of Prof. Alexadru Lupaş to obtai... 8 Valahia Uiversity,Târgovişte Departmet of Mathematics, 8 Bd. Uirii, 30082 Târgovişte E-mail address: averescu@clicket.ro