Several Answers to an Open Problem

Transcription

1 Int. J. Contemp. Mth. Sciences, Vol. 5, 2010, no. 37, Severl Answers to n Open Problem Xinkun Chi, Yonggng Zho nd Hongxi Du College of Mthemtics nd Informtion Science Henn Norml University Henn Province, , Chin chixinkun@gmil.com ztyg68@tom.com duhongxi24@gmil.com Abstrct In this pper, we estblish severl nswers to n open problem which is posed by Liu et l. in [5]. Mthemtics Subject Clssifiction: 26D15 Keywords: Qi-type inequlity; integrl inequlity; open problem 1 Introduction The following problem ws posed by Qi in his rticle [13]: Under wht condition does the inequlity ( [ ] b t 1 t f(x dx f (1 hold for t>1?. There re numerous nswers nd extension results to this open problem [1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 16]. These results were obtined by different pproches, such s, e.g. Jensen s inequlity, the convexity method [16]; functionl inequlities in bstrct spces [1, 2]; probbility mesures view [4, 7]; Hölder inequlity nd its reversed vrints [2, 12]; nlyticl methods [11, 15]; Cuchy s men vlue theorem [3, 14]. In [9], the uthors introduced the following discrete version of (1 s follows, Under wht condition does the inequlity ( n n β x α i i x i i (2 i=1 i=1

2 1814 X. K. Chi, Y. G. Zho nd H. X. Du hold for α, β > 0?. (For the infinite series, the sme method in the bove finite series cn be discussed. Very recently, some similr discrete inequlities were developed (for instnce, the reference [10]. In [5], the uthors posed the following open problem. Open Problem 1. Under wht conditions does the inequlity hold for α, β nd? ( f α+β (x α f β (3 In the pper, we will estblish severl nswers to the bove open problem. 2 Min results Theorem 2.1. Let f(x,g(x > 0 be continuous functions on [, b]. Assume tht α, β > 0 nd Then we hve ( g α+β 1 for ll x [, b]. (4 g α (xf β β ( f α+β. Proof. By the Hölder s inequlity nd the condition (4, we hve ( g α (xf β β ( ( α ( g α+β β b f α+β. f α+β From the bove theorem we hve the following Corollry 2.2. Let α, β > 0 nd (b +1 α + β +1. (5 Then ( (x α f β β ( (x α f β.

3 Severl nswers to n open problem 1815 Theorem 2.3. Let f(x,g(x > 0 be continuous functions on [, b]. (1 Assume tht 1, α 0,β R nd then we cn get ( (1 /α g (x f β f(x for ll x [, b], (6 ( ( g α (xf β f α+β. (7 (2 Assume tht <0, α 0,β R nd (6, then the inequlity (7 holds. (3 Assume tht 0 < 1, α 0,β R nd ( (1 /α g (x f β f(x for ll x [, b], (8 then the inequlity in (7 reverses. Proof. For ny function h(x, let E μ h(x = h(xμ(dx where μ(dx = f β f β. Then we hve ( ( f α+β = f β [E μ f α (X] nd by the Jesen s inequlity, ( ( g α (xf β = From the condition (6, we hve ( [ f β E μ g α (X ] ( which implies the desired result. f β [E μ g α (X] ( [ f β E μ g α (X ]. f β [E μ f α (X]

4 1816 X. K. Chi, Y. G. Zho nd H. X. Du Theorem 2.4. Let f(x,g(x > 0 be continuous functions on [, b]. Assume tht α 1, g M, nd then the inequlity (7 holds. ( α (b 1 M α f β 1, (9 Proof. Let X denote the uniform rndom vrible on [, b], then it is not difficult to check ( b gα (xf β f α+β which implies the desired result. References = (b ( Eg α (Xf β (X (b Ef α+β (b 1 M α ( Ef β (X Ef α+β (b 1 M α ( Ef β (X α [1] M. AKKOUCHI, On n integrl inequlity of Feng Qi, Divulg. Mt., 13(1 (2005, [2] L. BOUGOFFA, Notes on Qi type integrl inequlities, J. Inequl. Pure nd Appl. Mth., 4(4 (2003, Art. 77. [3] Y. CHEN nd J. KIMBALL, Note on n open problem of Feng Qi, J. Inequl. Pure nd Appl. Mth., 7(1 (2006, Art. 4. [4] V. CSISZÁR nd T.F. MÒRI, The convexity method of proving momenttype inequlities, Sttist. Probb. Lett., 66 (2004, [5] W. J. LIU, Q. A. NGÔ; nd V. N. HUY, Severl interesting integrl inequlities. J. Mth. Inequl., 3(2 (2009, [6] S. MAZOUZI nd F. QI, On n open problem regrding n integrl inequlity, J. Inequl. Pure nd Appl. Mth., 4(2 (2003, Art. 31. [7] Y. MIAO, Further development of Qi-type integrl inequlity, J. Inequl. Pure nd Appl. Mth., 7(4 (2006, Art [8] Y. MIAO nd J. F. LI, Further development of n open problem. J. Inequl. Pure nd Appl. Mth., 9(4 (2008, Art. 108.

5 Severl nswers to n open problem 1817 [9] Y. MIAO nd J. F. LIU, Discrete results of Qi-type inequlity. Bull. Koren Mth. Soc., 46(1 (2009, [10] Y. MIAO nd F. QI, A discrete version of n open problem nd severl nswers. J. Inequl. Pure nd Appl. Mth., 10(2 (2009, Art. 49. [11] J. PE CARIĆ nd T. PEJKOVIĆ, Note on Feng Qi s integrl inequlity, J. Inequl. Pure nd Appl. Mth., 5(3 (2004, Art. 51. [12] T. K. POGÁNY, On n open problem of F.Qi, J. Inequl. Pure nd Appl. Mth., 3(4 (2002, Art. 54. [13] F. QI, Severl integrl inequlities, J. Inequl. Pure nd Appl. Mth., 1(2 (2000, Art. 19. [14] F. QI, A. J. LI, W. Z. ZHAO, D.W. NIU nd J. CAO, Extensions of severl integrl inequlities, JIPAM. J. Inequl. Pure nd Appl. Mth., 7(3 (2006, Art [15] N. TOWGHI, Notes on integrl inequlities, RGMIA Res. Rep. Coll., 4(2 (2001, Art. 10, [16] K.-W. YU nd F. QI, A short note on n integrl inequlity, RGMIA Res. Rep. Coll., 4(1 (2001, Art. 4, Received: Mrch, 2010