On Two Integrability Methods of Improper Integrals

Transcription

1 Inernaional Journal of Mahemaics and Compuer Science, 13(218), no. 1, 45 5 M CS On Two Inegrabiliy Mehods of Improper Inegrals H. N. ÖZGEN Mahemaics Deparmen Faculy of Educaion Mersin Universiy, TR Mersin, Turkey nogduk@gmail.com (Received Sepember 28, 217, Acceped Ocober 2, 217) Absrac In his paper, we inroduce a new concep of N,p k,k 1 inegrabiliy of improper inegrals. By using his definiion we prove an analogous heorem due o Bor [H. Bor, On wo summabiliy mehods, Mah. Proc. Cambridge Philos. Soc., 97, 1985, ] for improper inegrals raher han infinie series. 1 Inroducion Throughou his paper we assume ha f is a real valued funcion which is coninuous on [, ) and s(x) = f()d. By σ(x), we denoe he Cesàro mean of s(x). The inegral f()d is said o be inegrable C,1 k,k 1, in he sense of Fle [4], if x k 1 σ (x) k = v(x) k x (1) is convergen. Here, v(x) = 1 x f()d. f()dis called he generaor of he inegral Key words and phrases: Cesàro and Riesz mehod, improper inegrals, divergen inegrals, inequaliies for inegrals, Hölder inequaliy. AMS (MOS) Subjec Classificaions: 26D15, 4F5, 4A1, 35A23. ISSN , 218, hp://ijmcs.fuure-in-ech.ne

2 46 H. N. ÖZGEN Le p be a real valued, non-decreasing funcion on [, ) such ha P(x) = The Riesz mean of s(x) is defined by σ p (x) = 1 P(x) p()d,,p() =. p()s()d. We say ha he inegral f()d is inegrable N,p k,k 1, if σ p (x) k dx (2) is convergen. In Paricular, if we ake = 1 for all values of x, hen N,p k inegrabiliy reduces o C,1 k inegrabiliy of improper inegrals. Given any funcions f,g, i is cusomary o wrie g(x) = O(f(x)), if here η. The difference beween s(x) and is nh weighed mean σ p (x), which is called he weighed Kronecker ideniy, is given by he ideniy exis η and N, for every x > N, g(x) f(x) where s(x) σ p (x) = v p (x), (3) v p (x) = 1 P(u)f(u)du. P(x) We noe ha if we ake = 1, for all values of x, hen we have he following ideniy(see [3]) s(x) σ(x) = v(x). Since σ p (x) = P(x) v, condiion (3) can be rewrien as s(x) = v p (x)+ p(u) P(u) v p(u)du. (4)

3 On Two Inegrabiliy Mehods of Improper Inegrals 47 In view of he ideniy (4), he funcion v p (x) is called he generaor funcion of s(x). The absolue Nörlund summabiliy of Fourier series and is allied series was sudied by several workers. The corresponding problem of absolue summabiliy of rigonomeric inegrals by he funcional Nörlund mehods up o he presen was no been sudied o he same degree. In fac, a paper of Bo icun [1] is he firs one which akes up he problem in a special direcion. Nex, in 1974, Lal [5] generalized Bo icun s resul. Lal and Ram [6], [7], Lal and Singh [8], [9] invesigaed several condiions for he absolue Nörlund summabiliy of inegrals associaed wih he Fourier inegral of a funcion. Recenly, he auhor ([1], [11]) esablished several heorems dealing wih he absolue Cesáro and Riesz inegrabiliy of improper inegrals, respecively. In wha follows, we give he main heorem. 2 Main resul The aim of his paper is o prove he heorem due o Bor [2] for improper inegrals. Now we can give he following heorem. Theorem 2.1. Le p be a real valued, non-decreasing funcions on [, ) such ha as x saisfying (2) and x = O(P(x)), (5) P(x) = O(x). (6) If f()d is inegrable C,1 k, hen i is also inegrable N,p k,k 1. 3 Proof of he Theorem Le σ p (x) denoe he ( N,p) means of he inegral f()d. Then we have σ p (x) = 1 P(x) p()s()d. (7) Since he inegral f()d is inegrable C,1 k, we can wrie v(x) k dx x

4 48 H. N. ÖZGEN is convergen, where v(x) is he generaor funcion of s(x). Differeniaing he equaion (7), we can wrie σ = P 2 (x) P()f()d = P 2 (x) P() f()d. Inegraing by pars of he second saemen, we obain σ p x ( ) P() (x) = v(x) v()d P 2 (x) P 2 (x) = x v(x) p()v()d+ P 2 (x) P 2 (x) P 2 (x) = σ p,1 (x)+σ p,2 (x)+σ p,3 (x), say. P() v()d To complee he proof of he heorem, i is sufficien o show ha σ p,r(x) k dx = O(1) as m, for r = 1,2,3. Using he condiion (5), we have ( ) k 1 P(x) m σ p,1(x) k dx = P(x) v(x) k dx 1 = x v(x) k dx = O(1) as m by virue of he hypoheses of he heorem.

5 On Two Inegrabiliy Mehods of Improper Inegrals 49 Applying Hölder s inequaliy wih k > 1, we ge m ( σ p,2(x) k dx x ) k dx = O(1) p() v() d P k+1 (x) ( dx x ) = O(1) p() v() k d P(x) ( 1 x ) k 1 p()d P(x) = O(1) p() v() k d P 2 (x) dx by virue of he hypoheses of he heorem. Finally, as in σ p,2 (x), by (6), we have ha m σ p,3(x) k dx = O(1) p() = O(1) P() v() k d v() k = O(1) d = O(1) as m ( dx x P k+1 (x) ( dx = O(1) P k+1 (x) = O(1) as m by virue of he hypoheses of he heorem. So, we ge σ p (x) k dx P() ) k v() d p() v() d is convergen. This complees he proof of he heorem. Therefore, a heorem of Bor [2] is obained for he improper inegrals. Acknowledgemen. The auhor acknowledges ha he resuls in his paper were presened a he Inernaional Workshop, Mahemaical Mehods in Engineering, MME-217, held in Cankaya Universiy, Ankara, Turkey on April 27-29, 217 and his work was suppored by Mersin Universiy Coordinaorship of Scienific Research Projecs. ) k

6 5 H. N. ÖZGEN References [1] L. G. Boicun, Absolue summabiliy of conjugae Fourier inegrals by he mehod of G. F. Voronoi, Izv. Vys s. U cebn. Zaved. Maemaika, 61(1967), no. 6, [2] H. Bor, On wo summabiliy mehods, Mah. Proc. Cambridge Philos. Soc. 97(1985), [3] İ.Çanak, Ü.Tour, ATauberianheoremforCesàrosummabiliy facors of inegrals, Appl. Mah. Le. 24(211), no. 3, [4] T. M. Fle, On an exension of absolue summabiliy and some heorems of Lilewood and Paley, Proc. London Mah. Soc., 7(1957), [5] S. N. Lal, On he absolue summabiliy of he allied inegral by a funcional Nörlund mehod, Proc. Amer. Mah. Soc., 42(1974), no. 1, [6] S. N. Lal, S. Ram, On he absolue summabiliy of a rigonomeric inegral by a funcional Nörlund mehod, Indian J. Pure Appl. Mah. 5(1974), no. 1, [7] S. N. Lal, S. Ram, The absolue summabiliy of infinie inegrals by funcional Nörlund mehods, Indian J. Mah., 17(1975), no. 2, [8] S. N. Lal, A. K. Singh, Absolue Nörlund summabiliy of Fourier inegrals, Indian J. Mah., 21(1979), no. 2, [9] S. N. Lal, K. N. Singh, Allied inegral and absolue funcional Nörlund mehod of summabiliy, Indian J. Pure Appl. Mah., 11(198), no. 12, [1] H. N. Özgen, On C,1 k inegrabiliy of Improper inegrals, In. J. Anal. Appl., 11(216), no. 1, [11] H. N. Özgen, On equivalence of wo inegrabiliy mehods, Miskolc. Mah. Noes, 18(217), no. 1,