Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Note on Saigo’s Fractional Integral Inequalities

Guotao Wang*, Harshvardhan Harsh, S.D. Purohit, Trilok Gupta

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, People’s Republic of China

Abstract: In this paper, some new integral inequalities related to the bounded functions, involving Saigo’s fractional integral operators, are eshtablished. Special cases of the main results are also pointed out.

Keywords: Integral inequalities, Gauss hypergeometric function, Saigo’s fractional integral operators

Received May 26, 2014; Revised June 09, 2014; Accepted June 12, 2014

Citation: Guotao Wang, Harshvardhan Harsh, S.D. Purohit, Trilok Gupta, A Note on Saigo’s Fractional Integral Inequalities. Turkish Journal of Analysis and Number Theory, 2 (3). 65-69. January 2014. doi:10.12691/tjant-2-3-2

1. Introduction and Preliminaries

Under various assumptions (Chebyshev inequality, Grüss inequality, Minkowski inequality, Hermite- Hadamard inequality, Ostrowski inequality etc.), inequalities are playing a very significant role in all fields of mathematics, particularly in the theory of approximations (see [2,6,7,13,14,17,23]). Therefore, in the literature we found several extensions and generalizations of these integral inequalities for the functions of bounded variation, synchronous, Lipschitzian, monotonic, absolutely continuous and n-times differentiable mappings etc. ([10,11,12,15,16,19,20,21,22,26,27,28]). In the past recent years, one more dimension have been added to this study, by introducing number of integral inequalities involving various fractional calculus and q-calculus operators. For detailed account, one may refer [1,3,4,5,8,9,18,24,25,29-35] and the references cited therein.

Recently, Tariboon et al. [33] investigated certain new integral inequalities for the integrable functions, whose bounds are also integrable functions, involving the Riemann-Liouville fractional integral operators. Our aim in this paper, is to obtain a general extensions of the results due to Tariboon et al. [33]. Main results investigated here provide certain new integral inequalities associated with the integrable functions, whose bounds are also integrable functions, involving the Saigo’s fractional integral operators. We also give some consequent results and special cases of the main results.

Firstly, we mention below the basic definitions and notations of some well-known operators of fractional calculus, which shall be used in the sequel.

Let then the Saigo fractional integral of order for a real-valued continuous function is defined by ([36], see also [[37], p. 19]):

[picture] | (1.1)

where, the function appearing as a kernel for the operator (1.1) is the Gaussian hypergeometric function defined by

[picture] | (1.2)

and is the Pochhammer symbol

The operator includes both the Riemann-Liouville and the Erdélyi-Kober fractional integral operators given by the following relationships:

[picture] | (1.3)

and

[picture] | (1.4)

Following [36], for in (1.1), we get

[picture] | (1.5)

2. Main Results

In this section, we obtain certain integral inequalities, related to the integrable functions, whose bounds are also integrable functions, involving Saigo’s fractional hypergeometric operators. The results are given in the form of the following theorems:

Theorem 1. Let and are integrable functions defined on such that

[picture] | (2.1)

Then, for we have

[picture] | (2.2)

where

Proof. By the hypothesis of inequality (2.1), for any we have

which follows that

[picture] | (2.3)

Consider

[picture] | (2.4)

which remains positive, for all under the conditions stated with Theorem 1. Multiplying both sides of (2.3) by (where is given by (2.4)) and integrating the resulting identity with respect to from 0 to , and using (1.1), we get

[picture] | (2.5)

Next, on multiplying both sides of (2.5) by

[picture] | (2.6)

which also remains positive, for all Upon integrating the resulting inequality so obtained with respect to from 0 to , and using the operator (1.1), we easily arrive at the desired result (2.1).

It may be noted that, for the Theorem 1 immediately reduces to the following result:

Corollary 1. Let and are integrable functions defined on and satisfying inequality (2.1). Then, for we have

[picture] | (2.7)

where

Theorem 2. Let and be two integrable functions on and and are four integrable functions on , such that

[picture] | (2.8)

Then, for the following inequalities holds true:

[picture] | (2.9)

[picture] | (2.10)

[picture] | (2.11)

[picture] | (2.12)

Proof. Let f and g are two integrable functions and satisfying inequality (2.8), then to prove (2.9), we can write

which follows that

[picture] | (2.13)

On multiplying both sides of (2.13) by (where is given by (2.4)) and integrating with respect to from 0 to , then by making use of (1.1), we get

[picture] | (2.14)

Next, multiplying both sides of (2.13) by where is given by (2.6), and integrating with respect to from 0 to , we easily arrive at the desired result (2.9).

Following the similar procedure, one can easily establish the remaining inequalities (2.10) to (2.12) by using the following inequalities, respectively

and

Therefore, we omit the further details of the proof of these results.

3. Consequent Results and Special Cases

The Saigo’s fractional integral operator defined by (1.1), possess the advantage that the Erdélyi-Kober and the Riemann-Liouville type fractional integral operators happen to be the particular cases of this operator. Therefore, by suitably specializing the parameters, we now briefly consider some special cases of the result derived in the preceding section. To this end, let us set and and make use of the relation (1.4), then Theorems 1 & 2 yields the following inequalities involving the Erdélyi-Kober type fractional integral operators:

Corollary 2. Let and are integrable functions defined on and satisfying inequality (2.1), then for we have

[picture] | (3.1)

where

Corollary 3. Let and be two integrable functions on and and are four integrable functions on and satisfying inequality (2.8). Then, for and the following inequalities holds true:

[picture] | (3.2)

[picture] | (3.3)

[picture] | (3.4)

[picture] | (3.5)

Next, if we replace by by and make use of the relation (1.3), then Theorems 1 & 2 corresponds to the known integral inequalities involving Riemann-Liouville type fractional integral operators, due to Tariboon et al [33].

Further, if we put and where and make use of formula (1.5), then the Theorems 1 & 2 leads to the following particluar results:

Corollary 4. Let f be an integrable function defined on , such that

[picture] | (3.6)

Then, for we have

[picture] | (3.7)

where and

Corollary 5. Let and be two integrable functions on such that

[picture] | (3.8)

Then, for and the following inequalities holds true:

[picture] | (3.9)

[picture] | (3.10)

[picture] | (3.11)

[picture] | (3.12)

Again, if we set and and make use of formula (1.5), then the Theorem 1 and Corollary 1, further leads to the following integral inequalities:

Corollary 6. Let f be an integrable function defined on , such that

Then, for and we have

[picture] | (3.13)

Corollary 7. Let be an integrable function defined on such that

Then, for we have

[picture] | (3.14)

In this paper, we have introduced certain general integral inequalities, related to the integrable and bounded functions and involving Saigo’s fractional integral operators. Therefore, we conclude with the remark that, by suitably specializing the arbitrary function and one can further easily obtain additional integral inequalities involving the Riemann-Liouville, Erd´elyi-Kober and Saigo type fractional integral operators from our main results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

References

[1]   | Anastassiou, G.A.: Advances on Fractional Inequalities, Springer Briefs in Mathematics, Springer, New York, 2011.
[2]   | Ahmadmir, N. and Ullah, R.: Some inequalities of Ostrowski and Gr¨uss type for triple integrals on time scales, Tamkang J. Math., 42(4) (2011), 415-426.
[3]   | Baleanu, D. and Purohit, S.D.: Chebyshev type integral inequalities involving the fractional hypergeometric operators, Abstract Appl. Anal., 2014, Article ID 609160, 10 (pp).
[4]   | Baleanu, D., Purohit, S.D. and Agarwal, P.: On fractional integral inequalities involving hypergeometric operators, Chinese Journal of Mathematics, 2014, Article ID 609476, 5(pp.).
[5]   | Belarbi, S. and Dahmani, Z.: On some new fractional integral inequalities, J. Inequal. Pure Appl. Math., 10(3)(2009), Art. 86, 5 (pp).
[6]   | Cerone, P. and Dragomir, S.S.: New upper and lower bounds for the Chebyshev functional, J. Inequal. Pure App. Math., 3 (2002), Article 77.
[7]   | Chebyshev, P.L.: Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98.
[8]   | Dahmani, Z. and Benzidane, A.: New weighted Gruss type inequalities via (α, β) fractional qintegral inequalities, International Journal of Innovation and Applied Studies, 1(1)(2012), 76-83.
[9]   | Dahmani, Z., Tabharit, L. and Taf, S.: New generalisations of Gr¨uss inequality using Riemann- Liouville fractional integrals, Bull. Math. Anal. Appl., 2 (3)(2010), 93-99.
[10]   | Dragomir, S.S.: A generalization of Grüss’s inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74-82.
[11]   | Dragomir, S.S.: A Grüss type inequality for sequences of vectors in inner product spaces and applications, J. Inequal. Pure Appl. Math., 1(2) (2000), 1-11.
[12]   | Dragomir, S.S.: Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 31(4)(2000), 397-415.
[13]   | Dragomir, S.S.: Operator Inequalities of the Jensen,Čebyšev and Grüss Type, Springer Briefs in Mathematics, Springer, New York, 2012.
[14]   | Dragomir, S.S. and Wang, S.: An inequality of Ostrowski-Grüss’ type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 13(11) (1997), 15-20.
[15]   | Gauchman, H.: Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281-300.
[16]   | Gavrea, B.: Improvement of some inequalities of Chebysev-Grüss type, Comput. Math. Appl., 64 (2012), 2003-2010.
[17]   | Grüss, D.: Uber das maximum des absoluten Betrages von Math. Z., 39(1935), 215-226.
[18]   | Kalla, S.L. and Rao, Alka: On Grüss type inequality for hypergeometric fractional integrals, Le Matematiche, 66 (1)(2011), 57-64.
[19]   | Kapoor, G.: On some discrete Gruss type inequalities, Int Jr. of Mathematical Sciences & Applications, 2(2) (2012), 729-734.
[20]   | Liu, Z.: Some Ostrowski-Grüss type inequalities and applications, Comput. Math. Appl., 53 (2007), 73-79.
[21]   | Maticć, M.: Improvment of some inequalities of Euler-Grüss type, Comput. Math. Appl., 46 (2003), 1325-1336.
[22]   | Mercer, McD A.: An improvement of the Grüss inequality, J. Inequa. Pure Appl. Math., 6(4) (2005), 1-4.
[23]   | Mitrinović, D.S., Pečarić, J.E. and Fink, A.M.: Classical and New Inequalities in Analysis, Kluwer Academic, 1993.
[24]   | Ntouyas, S.K., Purohit, S.D. and Tariboon, J.: Certain Chebyshev type integral inequalities involving the Hadamard’s fractional operators, Abstract Appl. Anal. 2014, Article ID 249091, 7(pp).
[25]   | Öğünmez, H. and Özkan, U.M.: Fractional quantum integral inequalities, J. Inequal. Appl., Volume 2011, Article ID 787939, 7 (pp).
[26]   | Özkan, U.M. and Yildirim, H.: Grüss type inequalities for double integrals on time scales, Comput. Math. Appl., 57 (2009), 436-444.
[27]   | Pachpatte, B.G.: On Grüss type integral inequalities, J. Inequa. Pure Appl. Math., 3 (1) (2002), 1-5.
[28]   | Pachpatte, B.G.: A note on Chebyshev-Grüss inequalities for differential equations, Tamsui Oxford Journal of Mathematical Sciences, 22(1), (2006), 29-36.
[29]   | Purohit, S.D. and Raina, R.K.: Chebyshev type inequalities for the Saigo fractional integrals and their q-analogues, J. Math. Inequal., 7(2) (2013), 239-249.
[30]   | Purohit, S.D. and Raina, R.K.: Certain fractional integral inequalities involving the Gauss hypergeometric function, Rev. Téc. Ing. Univ. Zulia, 37(2)(2014), In press.
[31]   | Purohit, S.D., U¸car, F. and Yadav, R.K.: On fractional integral inequalities and their q-analogues, Revista Tecnocientifica URU, 6 (2014), In press.
[32]   | Wang, G., Agarwal, P. and Chand, M.: Certain Grss type inequalities involving the generalized fractional integral operator. Journal of Inequalities and Applications 2014, 2014:147.
[33]   | Tariboon, J., Ntouyas, S.K. and Sudsutad, W.: Some new Riemann-Liouville fractional integral inequalities, Int. J. Math. Math. Sci., 2014, Article ID 869434, 6 (pp).
[34]   | Yang, W.: On weighted q-Čebyšev-Grüss type inequalities, Comput. Math. Appl., 61 (2011), 1342-1347.
[35]   | Zhu, C., Yang, W. and Zhao, Q.: Some new fractional q-integral Gr¨uss-type inequalities and other inequalities, J. Inequal. Appl., 2012 (2012), 299.
[36]   | Saigo, M.: A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ., 11 (1978) 135-143.
[37]   | Kiryakova, V.: Generalized Fractional Calculus and Applications (Pitman Res. Notes Math. Ser. 301), Longman Scientific & Technical, Harlow, 1994.