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Turkish Journal of Analysis and Number Theory

ISSN (Print): 2333-1100

ISSN (Online): 2333-1232

Website: http://www.sciepub.com/journal/TJANT

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Article

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions

1School of Advanced Sciences, VIT University Vellore - 632014, Tamilnadu, India


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 1-6
DOI: 10.12691/tjant-3-1-1
Copyright © 2015 Science and Education Publishing

Cite this paper:
G. Murugusundaramoorthy, T. Janani. An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions. Turkish Journal of Analysis and Number Theory. 2015; 3(1):1-6. doi: 10.12691/tjant-3-1-1.

Correspondence to: G.  Murugusundaramoorthy, School of Advanced Sciences, VIT University Vellore - 632014, Tamilnadu, India. Email: gmsmoorthy@yahoo.com

Abstract

The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out consequences of our main results.

Keywords

References

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Article

Common Fixed Point Results for Generalized Symmetric Meir-Keeler Contraction

1Department of Mathematics Govt. P. G. Arts & Science College Ratlam (M. P.) India


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 7-11
DOI: 10.12691/tjant-3-1-2
Copyright © 2015 Science and Education Publishing

Cite this paper:
Bhavana Deshpande, Amrish Handa. Common Fixed Point Results for Generalized Symmetric Meir-Keeler Contraction. Turkish Journal of Analysis and Number Theory. 2015; 3(1):7-11. doi: 10.12691/tjant-3-1-2.

Correspondence to: Bhavana  Deshpande, Department of Mathematics Govt. P. G. Arts & Science College Ratlam (M. P.) India. Email: bhavnadeshpande@yahoo.com

Abstract

We introduce the concept of generalized weakly compatibility for the pair {F,G} of mappings F,G:X×X→X and also introduce the concept of common fixed point of the mappings F,G:X×X→X. We establish a common fixed point theorem for generalized weakly compatible pair of mappings F,G:X×X→X without mixed monotone property of any mapping under generalized symmetric Meir-Keeler contraction on a non complete metric space, which is not partially ordered. An example supporting to our result has also been cited. We improve, extend and generalize several known results.

Keywords

References

[1]  V. Berinde, Coupled fixed point theorems for φ-contractive mixed monotone mappings in partially ordered metric spaces, Nonlinear Anal. 75 (2012), 3218-3228.
 
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[4]  L. Ciric, B. Damjanovic, M. Jleli and B. Samet, Coupled fixed point theorems for generalized Mizoguchi-Takahashi contractions with applications, Fixed Point Theory Appl. 2012, 2012:51.
 
[5]  B. Deshpande and A. Handa, Nonlinear mixed monotone-generalized contractions on partially ordered modified intuitionistic fuzzy metric spaces with application to integral equations, Afr. Mat.
 
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[6]  B. Deshpande and A. Handa, Application of coupled fixed point technique in solving integral equations on modified intuitionistic fuzzy metric spaces, Adv. Fuzzy Syst. Volume 2014, Article ID 348069, 11 pages. 10
 
[7]  H. S. Ding, L. Li and S. Radenovic, Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces, Fixed Point Theory Appl. 2012, 2012:96.
 
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[9]  N. Hussain, M. Abbas, A. Azam and J. Ahmad, Coupled coincidence point results for a generalized compatible pair with applications, Fixed Point Theory Appl. 2014, 2014: 62.
 
[10]  V. Lakshmikantham and L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (12) (2009), 4341-4349.
 
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[13]  B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72, 4508-4517 (2010).
 
[14]  B. Samet, E. Karapinar, H. Aydi and V. C. Rajic, Discussion on some coupled fixed point theorems, Fixed Point Theory Appl. 2013, 2013:50.
 
[15]  W. Sintunavarat, P. Kumam and Y. J. Cho, Coupled fixed point theorems for nonlinear contractions without mixed monotone property, Fixed Point Theory Appl. 2012, 2012: 170.
 
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Article

On the Numerical Regularity in the aspect of Prime Numbers

1Mathematical Society, Foundation Public School, Karachi, Pakistan


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 12-16
DOI: 10.12691/tjant-3-1-3
Copyright © 2015 Science and Education Publishing

Cite this paper:
Shaad P. Sufi. On the Numerical Regularity in the aspect of Prime Numbers. Turkish Journal of Analysis and Number Theory. 2015; 3(1):12-16. doi: 10.12691/tjant-3-1-3.

Correspondence to: Shaad  P. Sufi, Mathematical Society, Foundation Public School, Karachi, Pakistan. Email: shaadpyarali@gmail.com

Abstract

The purpose of this paper is to introduce a new pattern in Primes numbers, to eliminate the randomness in their patterns. This paper also justifies the solutions in a numerical and geometric manner. The Prime Function provides further distinction in the nature of Prime Numbers by distinguishing the nature of normality and Abnormality in Prime Numbers. To verify the normality of corresponding Prime numbers, the Sufi primality test is formed. Also using the Prime Function, the formula for the approximate sum of Prime Numbers is derived. The limitations and conditions of the Prime function are also stated. These factors provide a panoramic view of the Prime Function and its potential factor in Number Theory .

Keywords

References

[1]  https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford /Granville.pdf.
 
[2]  http://annals.math.princeton.edu/2014/179-3/p07.
 
[3]  http://en.wikipedia.org/wiki/Lucas_primality_test.
 

Article

Some Common Fixed Point Theorems for Weakly Contractive Maps in G-Metric Spaces

1Department of Mathematics Guru Jambheshwar University of Science and Technology, Hisar, India


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 17-20
DOI: 10.12691/tjant-3-1-4
Copyright © 2015 Science and Education Publishing

Cite this paper:
Manoj Kumar. Some Common Fixed Point Theorems for Weakly Contractive Maps in G-Metric Spaces. Turkish Journal of Analysis and Number Theory. 2015; 3(1):17-20. doi: 10.12691/tjant-3-1-4.

Correspondence to: Manoj  Kumar, Department of Mathematics Guru Jambheshwar University of Science and Technology, Hisar, India. Email: manojantil18@gmail.com

Abstract

In this paper, first we prove a common fixed point theorem for a pair of weakly compatible maps under weak contractive condition. Secondly, we prove common fixed point theorems for weakly compatible mappings along with E.A. and (CLRf) properties.

Keywords

References

[1]  Aamri M. and Moutawakil D. El., Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270, 181-188, 2002.
 
[2]  Aydi H., A fixed point result involving a generalized weakly contractive condition in G-metric spaces, Bulletin of Mathematical Analysis and Applications, vol. 3 Issue 4 (2011), 180-188.
 
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[7]  Mustafa Z., Obiedat H., and Awawdeh F., Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. Volume 2008, Article ID 189870, 12 pages, 2008.
 
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Article

On the Increasing Monotonicity of a Sequence Originating from Computation of the Probability of Intersecting between a Plane Couple and a Convex Body

1School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China

3Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 21-23
DOI: 10.12691/tjant-3-1-5
Copyright © 2015 Science and Education Publishing

Cite this paper:
BAI-NI GUO, FENG QI. On the Increasing Monotonicity of a Sequence Originating from Computation of the Probability of Intersecting between a Plane Couple and a Convex Body. Turkish Journal of Analysis and Number Theory. 2015; 3(1):21-23. doi: 10.12691/tjant-3-1-5.

Correspondence to: BAI-NI  GUO, School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China. Email: bai.ni.guo@gmail.com, bai.ni.guo@hotmail.com

Abstract

In the paper, the authors confirm the increasing monotonicity of a sequence which originates from the discussion on the probability of intersecting between a plane couple and a convex body.

Keywords

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[18]  R. L. Schilling, R. Song, and Z. Vondraˇcek, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012.
 
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Article

Fixed Point Results and E. A Property in Dislocated and Dislocated Quasi- metric Spaces

1Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Albania

2Department of Mathematics, KL University, Green Fields, Andhra Pradesh, India


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 24-29
DOI: 10.12691/tjant-3-1-6
Copyright © 2015 Science and Education Publishing

Cite this paper:
Kastriot Zoto, Arben Isufati, Panda Sumati Kumari. Fixed Point Results and E. A Property in Dislocated and Dislocated Quasi- metric Spaces. Turkish Journal of Analysis and Number Theory. 2015; 3(1):24-29. doi: 10.12691/tjant-3-1-6.

Correspondence to: Kastriot  Zoto, Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Albania. Email: zotokastriot@yahoo.com

Abstract

We prove several fixed points theorems for weakly compatible selfmappings on a dislocated and dislocated quasi-metric space, which satisfy E. A Like and common E. A. Like property, satisfying liner type of contractive condition.

Keywords

References

[1]  Aamri, M. and El Moutawakil, D. Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270, 181-188, 2002.
 
[2]  C. T. Aage and J. N. Salunke. The results on fixed points in dislocated and dislocated quasi-metric space. Appl. Math. Sci.,2(59):2941-2948, 2008.
 
[3]  F. M. Zeyada, G. H. Hassan, and M. A. Ahmed. A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces. The Arabian J. for Sci. and Eng., 31(1A): 111:114, 2005.
 
[4]  G. Jungck and B.E. Rhoades, Fixed points For Set Valued Functions without Continuity, Indian J. Pure Appl. Math., 29 (3) (1998), 227-238.
 
[5]  Liu, W., Wu, J. and Li, Z. Common fixed points of single-valued and multi-valued maps, Int.J. Math. Math. Sc. 19, 3045–3055, 2005.
 
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[6]  K. Zoto, Weakly compatible mappings and fixed points in dislocated -metric spaces, International journal of mathematical archive, vol. 4 (6), 2013, 131-137.
 
[7]  K. Zoto and E. Hoxha, Fixed point theorems in dislocated and dislocated quasi-metric spaces, Journal of Advanced Studies in Topology; Vol. 3, No.4, 2012.
 
[8]  K. Jha and D. Panthi, A Common Fixed Point Theorem in Dislocated Metric Space, Appl. Math. Sci., vol. 6, 2012, no. 91, 4497-4503.
 
[9]  K. P. R. Rao and P. Rangaswamy, Common Fixed Point Theorem for Four Mappings in Dislocated Quasi-Metric Space, The Nepali Math. Sci. Report, 30 (1-2), 2010, 70-75.
 
[10]  P. Hitzler and A. K. Seda. Dislocated topologies. J. Electr. Engin., 51(12/S):3:7, 2000.
 
[11]  P. S Kumari, Common fixed point theorems on weakly compatible maps on dislocated metric spaces, Mathematical Sciences 2012, 6:71.
 
[12]  R.Shrivastava, Z.K.Ansari and M.Sharma. Some results on Fixed Points in Dislocated and Dislocated Quasi-Metric Spaces. Journal of Advanced Studies in Topology; Vol. 3, No.1, 2012.
 
[13]  S. K. Vats, Weakly Compatible Maps in Metric Spaces, J. Indian Math. Soc., 69 (1-4), (2002), 139-143.
 
[14]  M. Arshad, A. Shoaib and P. Vetro; Common fixed points of a pair of Hardy Rogers type mappings on a closed ball in ordered dislocated metric spaces. Journal of function spaces and applications, vol 2013, article id 638181.
 
[15]  E. Karapinar and P. Salimi, Dislocated metric space to metric-like spaces with fixed point theorems. Fixed Point Theory and Applications 2013, 2013: 222.
 
[16]  M. Arshad, A. Shoaib and I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space. Fixed Point Theory and Applications 2013.
 
[17]  Yijie Ren,Junlei Li, and Yanrong Yu, Common fixed point theorems for nonlinear contractive mappings in dislocated metric spaces. Abstract and Applied Analysis vol 2013, article id 483059.
 
[18]  N. Hussain, J.R. Roshan, V. Parvaneh and M. Abbas; Common fixed point results for weak contractive mappings in ordered b-dislocated metric spaces with applications. Journal of Inequalities and Applications 2013, 2013:486.
 
[19]  P Sumati Kumariet al, New Version for Hardy and Rogers Type Mapping in Dislocated Metric Space. International Journal of Basic and Applied Sciences, 1 (4) (2012) 609-617.
 
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[21]  K. Wadhwa, H. Dubey, R. Jain; Impact of E. A. Like property on common fixed point theorems in fuzzy metric spaces.J. Adv. Stud. Topology 3 (1) (2012), 52-59.
 
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Article

Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative

1School of Mathematics and Allied Sciences, Jiwaji University, Gwalior

2Department of Mathematics RJIT, BSF Academy, Tekanpur, Gwalior

3Department of Mathematics, National Institute of Technology, Silchar - 788 010, District - Cachar (Assam), India

4L. 1627 Awadh Puri Colony Beniganj, Phase – III, Opposite – Industrial Training Institute (I.T.I.), Faizabad - 224 001 (Uttar Pradesh), India

5Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat – 395 007 (Gujarat), India


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 30-32
DOI: 10.12691/tjant-3-1-7
Copyright © 2015 Science and Education Publishing

Cite this paper:
Mohd. Farman Ali, Manoj Sharma, Lakshmi Narayan Mishra, Vishnu N. Mishra. Dirichlet Average of Generalized Miller-Ross Function and Fractional Derivative. Turkish Journal of Analysis and Number Theory. 2015; 3(1):30-32. doi: 10.12691/tjant-3-1-7.

Correspondence to: Vishnu  N. Mishra, Applied Mathematics and Humanities Department, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat – 395 007 (Gujarat), India. Email: vishnunarayanmishra@gmail.com

Abstract

The object of the present paper is to establish the results of single Dirichlet average of Generalized Miller-Ross Function, using Riemann-Liouville Fractional Integral. The Generalized Miller-Ross Function can be measured as a Dirichlet average and connected with fractional calculus. In this paper the solution comes in compact form of single Dirichlet average of Generalized Miller-Ross Function. The special cases of our results are same as earlier obtained by Saxena et al. [12], for single Dirichlet average of Generalized Miller-Ross Function.

Keywords

References

[1]  Carlson, B.C., Special Function of Applied Mathematics, Academic Press, New York, 1977.
 
[2]  Carlson, B.C., Appell’s function F4 as a double average, SIAM J.Math. Anal. 6 (1975), 960-965.
 
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[4]  Carlson, B.C., Dirichlet averages of x t log x, SIAM J.Math. Anal. 18(2) (1987), 550-565.
 
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[11]  Mathai, A.M. and Saxena,R.K., The H-Function with Applications in Stastistics and other Disciplines, Wiley Halsted, New York, 1978.
 
[12]  Saxena,R.K., Mathai,A.M and Haubold, H.J., Unified fractional kinetic equation and a fractional diffusion equation, J. Astrophysics and Space Science 209 (2004) , 299-310.
 
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Article

Moment Problem and Inverse Cauchy Problems for Heat Equation

1Penza State University, Penza, Russia


Turkish Journal of Analysis and Number Theory. 2015, 3(1), 33-36
DOI: 10.12691/tjant-3-1-8
Copyright © 2015 Science and Education Publishing

Cite this paper:
O. Yaremko, N. Yaremko, T. Eliseeva. Moment Problem and Inverse Cauchy Problems for Heat Equation. Turkish Journal of Analysis and Number Theory. 2015; 3(1):33-36. doi: 10.12691/tjant-3-1-8.

Correspondence to: N.  Yaremko, Penza State University, Penza, Russia. Email: yaremki@mail.ru

Abstract

The solution of Hamburger and Stieltjes moment problem can be thought of as the solution of a certain inverse Cauchy problem. The solution of the inverse Cauchy problem for heat equation is founded in the form of Hermite polynomial series. The author reveals, the formulas obtained by him for the solution of inverse Cauchy problem have a symmetry with respect to the formulas for corresponding direct Cauchy problem. Obtained formulas for solution of the inverse problems can serve as a basis for the solution of the moment problem.

Keywords

References

[1]  Akhiezer, N.I., Krein, M.G. Some Questions in the Theory of Moments, Amer. Math. Soc., 1962.
 
[2]  Alifanov, O.M., Inverse problems of heat exchange, M, 1988, p. 279.
 
[3]  Arfken, G. B.; Weber, H. J. (2000), Mathematical Methods for Physicists (5th ed.), Boston, MA: Academic Press.
 
[4]  Bavrin, I. I., Yaremko, O. E. Transformation Operators and Boundary Value Problems in the Theory of Harmonic and Biharmonic Functions (2003) Doklady Mathematics, 68 (3), pp. 371-375.
 
[5]  Baker, G. A., Jr.; and Graves-Morris, P. Padé Approximants. Cambridge U.P., 1996.
 
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[7]  Chebysev, P. Sur les valeurs limites des intégrales, Journal de Mathématiques pures et appliquées, 19 ( 1874), 157-160.
 
[8]  Krein, M.G. and Nudelman, A.A. The Markov Moment Problem and Extermal Problems, Translations of Mathematical Monographs, Volume Fifty, Library of Congress Cataloging in Publication Data, 1977.
 
[9]  Lavrentev, M.M., Some ill-posed problems of mathematical physics, Novosibirsk, AN SSSR, 1962, p. 92.
 
[10]  Mors, F.M., Fishbah, G. Methods of theoretical physics, 1958.
 
[11]  Yaremko, O.E. Matrix integral Fourier transforms for problems with discontinuous coefficients and transformation operators (2007) Doklady Mathematics, 76 (12), pp. 323-325.
 
[12]  Yaremko, O.E. Transformation operator and boundary value problems Differential Equation. Vol.40, No. 8, 2004, pp.1149-1160.
 
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Article

The Rogers-Ramanujan Identities

1Department of Mathematics, University of Chittagong, Bangladesh

2Department of Mathematics, Raozan University College, Bangladesh

3Premier University, Chittagong, Bangladesh


Turkish Journal of Analysis and Number Theory. 2015, 3(2), 37-42
DOI: 10.12691/tjant-3-2-1
Copyright © 2015 Science and Education Publishing

Cite this paper:
Fazlee Hossain, Sabuj Das, Haradhan Kumar Mohajan. The Rogers-Ramanujan Identities. Turkish Journal of Analysis and Number Theory. 2015; 3(2):37-42. doi: 10.12691/tjant-3-2-1.

Correspondence to: Haradhan  Kumar Mohajan, Premier University, Chittagong, Bangladesh. Email: haradhan1971@gmail.com

Abstract

In 1894, Rogers found the two identities for the first time. In 1913, Ramanujan found the two identities later and then the two identities are known as The Rogers-Ramanujan Identities. In 1982, Baxter used the two identities in solving the Hard Hexagon Model in Statistical Mechanics. In 1829 Jacobi proved his triple product identity; it is used in proving The Rogers-Ramanujan Identities. In 1921, Ramanujan used Jacobi’s triple product identity in proving his famous partition congruences. This paper shows how to generate the generating function for C'(n), C1'(n), C''(n), and C1''(n), and shows how to prove the Corollaries 1 and 2 with the help of Jacobi’s triple product identity. This paper shows how to prove the Remark 3 with the help of various auxiliary functions and shows how to prove The Rogers-Ramanujan Identities with help of Ramanujan’s device of the introduction of a second parameter a.

Keywords

References

[1]  Andrews, G.E, “An Introduction to Ramanujan’s Lost Notebook”, American Mathmatical Monthly, 86: 89-108. 1979.
 
[2]  Hardy, G.H. and Wright, E.M. “Introduction to the Theory of Numbers”, 4th Edition, Oxford, Clarendon Press, 1965.
 
[3]  Jacobi, C.G.J. (1829), “Fundamenta Nova Theoriae Functionum Ellipticarum (in Latin), Konigsberg Borntraeger, Cambridge University Press, 2012.
 
[4]  Baxter, R.J., “Exactly Solved Model in Statistical Models”, London, Academic Press, 1982.
 
[5]  Ramanujan, S., “Congruence Properties of Partitions”, Math, Z. 9: 147-153. 1921.
 
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[6]  Ramanujan, S., “Some Properties of P(n), Number of Partitions of n”, Proc. of the Cam. Philo. Society XIX, 207-210. 1919.
 
[7]  Das, S. and Mohajan, H.K., “Generating Function for P(n,p,*) and P(n, *,p)”, Amer. Rev. of Math. and Sta. 2(1): 33-35. 2014.
 
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Article

Some Generalizations of Integral Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China

2School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China


Turkish Journal of Analysis and Number Theory. 2015, 3(2), 43-48
DOI: 10.12691/tjant-3-2-2
Copyright © 2015 Science and Education Publishing

Cite this paper:
Tian-Yu Zhang, Bai-Ni Guo. Some Generalizations of Integral Inequalities of Hermite-Hadamard Type for n-Time Differentiable Functions. Turkish Journal of Analysis and Number Theory. 2015; 3(2):43-48. doi: 10.12691/tjant-3-2-2.

Correspondence to: Bai-Ni  Guo, School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China. Email: bai.ni.guo@hotmail.com

Abstract

In the paper, by establishing two integral identities and Hölder integral inequality, the authors generalize some integral inequalities of Hermite-Hadamard type for n-time differentiable functions on a closed interval.

Keywords

References

[1]  R.-F. Bai, F. Qi, and B.-Y. Xi, Hermite-Hadamard type inequalities for the m- and -logarithmically convex functions, Filomat 27(2013), no. 1, 1-7.
 
[2]  P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequality point of view, Handbook of Analytic-computational Methods in Applied Mathematics, 135-200, Chapman & Hall/CRC, Boca Raton, FL, 2000.
 
[3]  S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000; Available online at http://rgmia.org/monographs/hermite_hadamard.html.
 
[4]  C. P. Niculescu and L.-E. Persson, Convex Functions and their Applications, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23. Springer, New York, 2006.
 
[5]  F. Qi, Z.-L. Wei, and Q. Yang, Generalizations and refinements of Hermite-Hadamard's inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235-251.
 
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[6]  N. Ujević, Some double integral inequalities and applications, Acta Math. Univ. Comenian. (N.S.) 71 (2002), no. 2, 189-199.
 
[7]  B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex functions and applica-tions, Hacet. J. Math. Stat. 42 (2013), no. 3, 243-257.
 
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