Fudametal Joural of Mathematics ad Mathematical Scieces Vol. 7, Issue, 207, Pages 5-58 This paper is available olie at http://www.frdit.com/ Published olie March 2, 207 UNIFORMLY CONVERGENT STOLZ THEOREM FOR SEQUENCES OF FUNCTIONS CAO HAIJUN ad YAN HAN * Sciece College Shadog Jiaotog Uiversity P. R. Chia Shadog Uiversity of Fiace ad Ecoomics Shadog 250000 P. R. Chia Abstract Based o the lemma from Toplitz trasformatio for sequeces of uiformly coverget fuctios, we derive ad prove the uiformly coverget Stolz theorem for sequeces of fuctios i regio I.. Itroductio I this paper, we start from Stolz theorem for ifiite sequeces ad geeralize it to Stolz theorem for sequeces of fuctios. First we give the followig theorems for Keywords ad phrases: sequeces of fuctios, Toplitz trasformatio, Stolz theorem. 200 Mathematics Subect Classificatio: 40A30. This work is supported by a proect of Shadog Provice Higher Educatioal Sciece ad Techology Program (No. J4LI57); The Scietific Research Foudatio of Shadog Jiaotog Uiversity (No. Z20428), ad Research Fud for the Doctoral Program of Shadog Jiaotog Uiversity. * Correspodig author Received August 3, 206; Revised November 6, 206; Accepted November 24, 206 207 Fudametal Research ad Developmet Iteratioal
52 CAO HAIJUN ad YAN HAN ifiite sequeces: that If Theorem (Stolz Theorem [-5]). Let { a } ad { b } be two sequeces such () { a } is a strictly mootoe icreasig sequece; (2) lim a = +. b+ b lim = l exists (havig a limit of positive ad egative ifiity), the a a + b lim = l. a There is o requiremet for sequece { b } to have a limit of, so this theorem could be remarked as * Stolz theorem. Similarly, we geeralize Stolz theorem for sequeces of uiformly coverget fuctios with idetermiate expressio as followig: Theorem 2. If two sequeces of fuctios { f } ad { g } satisfy the followig coditios i regio I, () g is uiformly coverget to + o regio I; (2) For ay 0 < { g } mootoe icreases; (3) For ay give, fuctio f g + + f g is bouded o regio I. The, if f g + + f g is uiformly coverget to h, f g is uiformly coverget to h. Similar to the proof of Stolz theorem for ulimited sequeces, we eed to give Toplitz trasformatio for sequeces of fuctios which are the followig three lemmas. Lemma. Let sequeces of fuctios { f }, { g }, ad { a i } be
UNIFORMLY CONVERGENT STOLZ THEOREM FOR SEQUENCES 53 defied i regio I, if for ay : ( ) a 0 0 g x ( ) ( ) 0 ( ) a2 x a22 x g2 x = ( ) g x a a 2 a f f2, f ad the followig hold; ifiity; () 0, < i, i,, 2, L, x I; a i i (2) a =, i N, x I; i (3) For ay N, x I; a is uiformly coverget to 0 whe approaches (4) For ay, f is bouded i regio I. The, if { f } is uiformly coverget to h, a f = g h,, Proof. For ay, f is bouded i regio I, if { f } is uiformly coverget to h, the { f } ad h are uiformly bouded i regio I, that meas there exists M > 0 such that for ay, f M. Further because { f h} is uiformly coverget to 0, the for =, there exists N > 0, for ay > N, { f h} <. Let M = max{ M, M 2,, M N, }, where M i is the boud for f i h i regio I, thus for ay,
54 CAO HAIJUN ad YAN HAN f h M. As { f } is uiformly coverget to h, we kow that for ay > 0, there exists a N 0, for ay >, f h <. 2 Moreover, for ay, { a } is uiformly coverget to 0, N 0 M a uiformly coverget to 0, therefore for the above, 2 there exists a N, for ay > N, M a <. 2 Let N = N 0 + N, whe > N, for ay : g h = a f h = a f h a So = a f h a ( ) f h = a f h + a f h = + M a = + + a + =. 2 2 2 a f = g h,,
UNIFORMLY CONVERGENT STOLZ THEOREM FOR SEQUENCES 55 Thus Lemma holds. Lemma 2. Let sequeces of fuctios { f }, { g }, ad { a i } defied i regio I, for ay g = a f exists ad followig hold; k; () There exists k > 0, for ay N, a + a2 +... + a (2) For ay N, a is uiformly coverget to 0, ; (3) A = a + a2 +... + a is uiformly coverget to as, x I; (4) For ay, f is bouded i regio I. The, if { f } is uiformly coverget to 0,, a f = g 0,, Proof. Sice f is uiformly coverget to 0, ad for ay, f is bouded i regio I, we kow that { f } is uiformly bouded i regio I, which meas that there exists M > 0, for ay N, { a } uiformly coverget to 0, ad f M. Similarly, for ay > 0, there exists a N, for ay >, f <. Sice for ay, a is uiformly 2k coverget to 0, further we kow M a is uiformly coverget to 0, thus for the above > 0, there exists a N > 0, for ay > N, x I 0 M a. 2 Let N = N 0 + N, whe > N, for ay
56 CAO HAIJUN ad YAN HAN g = a f a f + a f Thus Lemma 2 holds. N M a = + = + + a < + k =. 2k 2 2k Lemma 3. Let sequeces of fuctios { f }, { g }, ad { a i } be N defied i regio I. If for ay g = a f followig hold: ad () There exists k > 0, for ay N, a + + + a ; a2 k (2) For ay N, { a } is uiformly coverget to 0, ; (3) A = a + a2 + + a is uiformly coverget to as, x I; (4) For ay, f is bouded i regio I. The, if { f } is uiformly coverget to h, ca get a f = g h,, Proof. Let A = + α, the { α } is uiformly coverget to 0, we g h = a f [ A α] h = = a ( f h) + a2( f2 h)
UNIFORMLY CONVERGENT STOLZ THEOREM FOR SEQUENCES 57 + + a( f h) + h α = Z + h α. Sice { f } is uiformly coverget to h, we ca coclude that { f h} is uiformly coverget to 0, hece { Z } is uiformly coverget to 0, the g h = Z + h α. As for ay, f is bouded i regio I, we kow that h is bouded, which meas there exists a M > 0 such that for ay h M. O the other had, α is uiformly coverget to 0, so, for ay > 0, there exists a N > 0 such that for ay > N, α <, h α M α < M. Thus { h α } is uiformly coverget to 0, furthermore, we ca coclude that { g h} is uiformly coverget to 0, ad { g } is uiformly coverget to h. Thus Lemma 3 holds. Now we use the above lemmas to prove Theorem 2. f+ f Proof. Let z = g+ g g g a =, the g be uiformly coverget to h ad w = a z + a2 z2 + + a z = f g f0 g. Sice { g } is uiformly coverget to +, so = g g is uiformly coverget to 0 (, x I ), the g a A = a + a2 + + a = g g0 g
58 CAO HAIJUN ad YAN HAN is uiformly coverget to. The from above Lemma 3, w = a z h,, f0 Moreover, sice ( ) is uiformly coverget to 0, that g x uiformly coverget to h,, Fially, Theorem 2 holds. f g ( ) x is Refereces [] Mathematical Aalysis, Vol., Mathematics Departmet, Huadog Normal Uiversity, Chia Higher Educatio Press, Beiig, 2008, pp. 78-98. [2] Liu Lisha, Basic theory ad typical methods for mathematical aalysis, Chia Sciece ad Techology Press, Beiig. [3] W. Flemig, Multivariate Fuctio, Traslated by Zhuag Yadog, People s Educatio Press, Beiig, 982, pp. 94-99. [4] Ouyag Guagzhog, Zhu Xueya, Ji Fuli ad Zhu Chuazhag, Mathematical aalysis, Chia Higher Educatio Press, Beiig, 2007, pp. 84-93. [5] Liu Yulia ad Fu Peire, Mathematical aalysis lecture, 3rd ed., Chia Higher Educatio Press, Beiig, 992.