ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II
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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn Sălăgen on his 60 th birthdy Abstrct. The pper dels with the symptotic behvior of the intermedite points in the men vlue theorems for integrls s the involved intervl shrinks to zero. 1. Introduction Especilly in the lst two decdes gret del of work hs been done in connection with the symptotic behvior of intermedite points in certin men vlue theorems (see, for instnce, [1], [2], [3], [5], [9], [12], [13], [14]). The investigtions in this direction strted with the pper by Azpeiti [3], deling with the symptotic behvior of the intermedite point in the Lgrnge-Tylor men vlue theorem. A significnt step forwrd ws relized by Abel [1], who obtined complete symptotic expnsion of the intermedite point in the Lgrnge-Tylor men vlue theorem when the length of the involved intervl pproches zero. Lter, following Abel s method of proof, similr complete symptotic expnsions hve been obtined by severl uthors for other men vlue theorems (Abel nd Ivn [2] for the differentil men vlue theorem of divided differences, Xu, Cui nd Hu [13] for the differentil men vlue theorem of divided differences with repetitions, Trif [12] for the Pwlikowsk men vlue theorem). The purpose of the present pper is to continue our investigtions strted in [12]. But unlike the pper [12], here we del with the symptotic behvior of the Received by the editors: Mthemtics Subject Clssifiction. 26A06. Key words nd phrses. men vlue theorems for integrls, symptotic pproximtions. 241
2 TIBERIU TRIF intermedite points in the men vlue theorems for integrls s the involved intervl shrinks to zero. theorems for integrls. For the reder s convenience we recll first the two men vlue Theorem 1.1 (first men vlue theorem for integrls). If f : [, b] R is continuous function nd g : [, b] [0, ) is nonnegtive Riemnn integrble function, then there is number c [, b] such tht f(t)g(t)dt = f(c) Corollry 1.2. If f : [, b] R is continuous function, then there is number c [, b] such tht f(t)dt = f(c)(b ). Theorem 1.3 (second men vlue theorem for integrls). If f : [, b] R is monotone nd g : [, b] R is Riemnn integrble on [, b], then there is number c [, b] such tht f(t)g(t)dt = f() c g(t)dt + f(b) c The second men vlue theorem for integrls is instrumentl in theories like trigonometric series or Lplce trnsforms (see [8] for proof nd [11] for n interesting ppliction of Theorem 1.3). If x (, b), then Theorem 1.1, Corollry 1.2 nd Theorem 1.3 pplied to the intervl [, x] insted of [, b] yield the existence of numbers [, b] s functions of x on (, b) such tht nd respectively. 242 f(t)g(t)dt = f( ) g(t)dt, (1.1) f(t)dt = f( )(x ), (1.2) cx f(t)g(t)dt = f() g(t)dt + f(x) g(t)dt, (1.3) Zhng [14, Theorem 4] proved tht the point in (1.2) stisfies lim x x = 1 n, (1.4) n + 1
3 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS provided tht f is continuous on [, b] nd n times differentible t with f (j) () = 0 (1 j n 1) nd f (n) () 0. In the specil cse when n = 1, n erlier result obtined by Jcobson [7] is recovered. In section 2 of our pper we obtin formul which is similr to (1.4), but involves the symptotic behvior of the point in the men vlue formul (1.1). The symptotic behvior of the point in the men vlue formul (1.3) is investigted in section Asymptotic behvior of the intermedite point in the first men vlue theorem for integrls In the proofs of the min results in this nd the next section we need the following Lemm 2.1. If p is nonnegtive integer nd ω : [, b] R is continuous function such tht ω(t) 0 s t, then ω(t)(t ) p dt = o((x ) p+1 ) (x ). Proof. Indeed, for every x (, b) by Theorem 1.1 there exists [, x] such tht ω(t)(t ) p dt = ω( ) (t ) p dt = ω() p + 1 (x )p+1. Since ω( ) 0 s x, we obtin the conclusion. Theorem 2.2. Suppose tht f, g : [, b] R re two functions stisfying the following conditions: (i) f is continuous on [, b] nd there is positive integer n such tht f is n times differentible t with f (j) () = 0 for 1 j n 1 nd f (n) () 0; (ii) g is nonnegtive, Riemnn integrble on [, b] nd there is nonnegtive integer k such tht g is k times differentible t with g (j) () = 0 for 0 j k 1 nd g (k) ()
4 TIBERIU TRIF Then the point in (1.1) stisfies k + 1 lim x x = n n + k + 1. (2.1) Proof. Without loosing the generlity we my ssume tht f() = 0. Indeed, otherwise we replce f by the function t [, b] f(t) f(). Note tht if stisfies (1.1), then stisfies lso (f(t) f())g(t)dt = (f( ) f()) By the Tylor expnsions of f nd g we hve f(t) = f (n) () (t ) n + ω(t)(t ) n, n! g(t) = g(k) () (t ) k + ε(t)(t ) k, k! where ω nd ε re continuous functions on [, b] stisfying ω(t) 0 nd ε(t) 0 s t. Therefore we hve f(t)g(t) = f (n) ()g (k) () n! k! (t ) n+k + γ(t)(t ) n+k, where γ is continuous on [, b] nd γ(t) 0 s t. By Lemm 2.1 we deduce tht f(t)g(t)dt = s x. By Lemm 2.1 we hve lso f (n) ()g (k) () n! k! (n + k + 1) (x )n+k+1 + o((x ) n+k+1 ) (2.2) Since g(t)dt = g(k) () (k + 1)! (x )k+1 + o((x ) k+1 ) (x ). f( ) = f (n) () n! nd 0 x, it follows tht 244 f( ) g(t)dt = f (n) ()g (k) () ( ) n + ω( )( ) n (x ) k+1 ( ) n + o((x ) n+k+1 ) (2.3)
5 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS s x. By (1.1), (2.2) nd (2.3) we conclude tht f (n) ()g (k) () (x ) k+1 ( ) n = f (n) ()g (k) () n! k! (n + k + 1) (x )n+k+1 + o((x ) n+k+1 ) (x ). Multiplying both sides by (x ) (n+k+1) /(f (n) ()g (k) ()) we get ( ) n cx = k + 1 x n + k o(1) (x ), whence the conclusion (2.1). Note tht if g(t) = 1 for ll t [, b], then (ii) is stisfied for k = 0. In this cse (1.1) becomes (1.2) nd (2.1) becomes (1.4), i.e., we recover Zhng s result mentioned in the introduction s specil cse of Theorem Asymptotic behvior of the intermedite point in the second men vlue theorem for integrls Theorem 3.1. Suppose tht f, g : [, b] R re two functions stisfying the following conditions: (i) f is monotone nd there is positive integer n such tht f is n times differentible t with f (j) () = 0 for 1 j n 1 nd f (n) () 0; (ii) g is Riemnn integrble on [, b] nd there is nonnegtive integer k such tht g is k times differentible t with g (j) () = 0 for 0 j k 1 nd g (k) () 0. Then the point in (1.3) stisfies Proof. Note tht (1.3) is equivlent to n lim x x = k+1 n + k + 1. (f(t) f())g(t)dt = (f(x) f()) 245
6 TIBERIU TRIF So, without loosing the generlity we my ssume tht f() = 0 (otherwise we replce f by the function t [, b] f(t) f()). Under the ssumption tht f() = 0 equlity (1.3) becomes f(t)g(t)dt = f(x) (3.1) By using the Tylor expnsions of f nd g nd proceeding s in the proof of Theorem 2.2 we deduce tht (2.2) holds nd tht f(x) g(t)dt = f (n) ()g (k) () By (3.1), (2.2) nd (3.2) we conclude tht f (n) ()g (k) () (x ) n [ (x ) k+1 ( ) k+1] (3.2) +o((x ) n+k+1 ) (x ) n [ (x ) k+1 ( ) k+1] (x ). = f (n) ()g (k) () n! k! (n + k + 1) (x )n+k+1 + o((x ) n+k+1 ) (x ). Multiplying both sides by (x ) (n+k+1) /(f (n) ()g (k) ()) we get 1 whence the conclusion. References ( cx x ) k+1 = k + 1 n + k o(1) (x ), [1] Abel, U., On the Lgrnge reminder of the Tylor formul, Amer. Mth. Monthly, 110 (2003), [2] Abel, U., Ivn, M., The differentil men vlue of divided differences, J. Mth. Anl. Appl., 325 (2007), [3] Azpeiti, A. G., On the Lgrnge reminder of the Tylor formul, Amer. Mth. Monthly, 89 (1982), [4] Duc, D. I., Properties of the intermedite point from the Tylor s theorem, Mth. Inequl. Appl., 12 (2009), [5] Duc, D. I., Pop, O., On the intermedite point in Cuchy s men-vlue theorem, Mth. Inequl. Appl., 9 (2006), [6] Duc, D. I., Pop, O. T., Concerning the intermedite point in the men vlue theorem, Mth. Inequl. Appl., 12 (2009),
7 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS [7] Jcobson, B., On the men vlue theorem for integrls, Amer Mth. Monthly, 89 (1982), [8] Porter, M. B., The second men vlue theorem for summble functions, Bull. Amer. Mth. Soc., 29(1923), no. 9, [9] Powers, R. C., Riedel, T., Shoo, P. K., Limit properties of differentil men vlues, J. Mth. Anl. Appl., 227 (1998), [10] Shoo, P. K., Riedel, T., Men Vlue Theorems nd Functionl Equtions, World Scientific, River Edge, NJ, [11] Strk, E. L., Appliction of men vlue theorem for integrls to series summtion, Amer. Mth. Monthly, 85 (1978), [12] Trif, T., Asymptotic behvior of intermedite points in certin men vlue theorems, J. Mth. Inequl., 2 (2008), [13] Xu, A., Cui F., nd Hu, Z., Asymptotic behvior of intermedite points in the differentil men vlue theorem of divided differences with repetitions, J. Mth. Anl. Appl., 365 (2010), [14] Zhng, B., A note on the men vlue theorem for integrls, Amer. Mth. Monthly, 104 (1997), Bbeş-Bolyi University Fculty of Mthemtics nd Computer Science Str. Kogălnicenu No. 1 RO Cluj-Npoc, Romni E-mil ddress: ttrif@mth.ubbcluj.ro 247
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