You are here

### Content: Volume 2, Issue 3

### Article

**On the Moments of the Function E*(t)**

^{1}Aleksandar Ivić, Katedra Matematike RGF-A, Universitet U Beogradu, Ðušina, Beograd, Serbia

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 102-109

**DOI:**10.12691/tjant-2-3-9

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Aleksandar Ivić. On the Moments of the Function E*(t).

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):102-109. doi: 10.12691/tjant-2-3-9.

Correspondence to: Aleksandar Ivić, Aleksandar Ivić, Katedra Matematike RGF-A, Universitet U Beogradu, Ðušina, Beograd, Serbia. Email: ivic@rgf.bg.ac.rs, aivic2000@yahoo.com

### Abstract

*the error term in the asymptotic formula for the mean square of If*

*with then we discuss bounds for third, fourth and fifth power moment of*

*We also prove that*

*always changes sign in for*

*and obtain (conditionally) the existence of its large positive, or small negative values.*

### Keywords

### References

[1] | F.V. Atkinson, The mean value of the Riemann zeta-function, Acta Math. 81(1949), 353-376. | ||

[2] | S.W. Graham and G. Kolesnik, Van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series 126, Cambridge University Press, Cambridge, 1991. vi+120 pp. | ||

[3] | A. Ivić, Large values of the error term in the divisor problem, Inventiones Math. 71(1983), 513-520. | ||

[4] | A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York, 1985 (2nd ed. Dover, Mineola, New York, 2003). | ||

[5] | A. Ivić, The mean values of the Riemann zeta-function, LNs 82, Tata Inst. of Fundamental Research, Bombay (distr. by Springer Verlag, Berlin etc.), 1991. | ||

[6] | A. Ivić, On the Riemann zeta-function and the divisor problem, Central European J. Math. (2)(4) (2004), 1-15; II, ibid. (3)(2) (2005), 203-214; III, Annales Univ. Sci. Budapest, Sect. Comp. 29(2008), 3-23.; IV, Uniform Distribution Theory 1(2006), 125-135. | ||

[7] | A. Ivić, On the mean square of the zeta-function and the divisor problem, Annales Acad. Scien. Fennicae Mathematica 23(2007), 1-9. | ||

[8] | A. Ivić, Some remarks on the moments of |(ζ(1/2+ it)| in short intervals, Acta Math. Hung. 119(2008), 15-24. | ||

[9] | A. Ivić, On some mean square estimates for the zeta-function in short intervals, Annales Univ. Sci. Budapest., Sect. Comp. 40(2013), 321-335. | ||

[10] | A. Ivić, On some mean value results for the zeta-function in short intervals, Acta Arith. 162.2(2014), 141-158. | ||

[11] | M. Jutila, Riemann’s zeta-function and the divisor problem, Arkiv Mat. 21(1983), 75-96 and II, ibid. 31(1993), 61-70. | ||

[12] | M. Jutila, On a formula of Atkinson, in “Coll. Math. Sci. J´anos Bolyai 34, Topics in classical Number Theory, Budapest 1981”, North-Holland, Amsterdam, 1984, pp. 807-823. | ||

[13] | T. Meurman, A generalization of Atkinson’s formula to L-functions, Acta Arith. 47(1986), 351-370. | ||

[14] | O. Robert and P. Sargos, Three-dimensional exponential sums with monomials, J. reine angew. Math. 591(2006), 1-20. | ||

[15] | K.-M. Tsang, Counting lattice points in the sphere, Bull. London Math. Soc. 32(2000), 679-688. | ||

### Article

**Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions**

^{1}Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, China

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

^{3}Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, China;Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 96-101

**DOI:**10.12691/tjant-2-3-8

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Li Yin, Li-Guo Huang, and Feng Qi. Some Inequalities for the Generalized Trigonometric and Hyperbolic Functions.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):96-101. doi: 10.12691/tjant-2-3-8.

Correspondence to: Li Yin, Department of Mathematics, Binzhou University, Binzhou City, Shandong Province, China. Email: yinli_79@163.com

### Abstract

### Keywords

### References

[1] | Á. Baricz, B. A. Bhayo, and M. Vuorinen, Turán type inequalities for generalized inverse trigonometric functions, available online at http://arxiv.org/abs/1305.0938. | ||

[2] | B. A. Bhayo and M. Vuorinen, Inequalities for eigen-functions of the p-Laplacian, available online at http: //arxiv.org/abs/1101.3911. | ||

[3] | B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no. 10, 1415-1426. | ||

[4] | B. A. Bhayo and M. Vuorinen, Power mean inequality of generalized trigonometric functions, available online at http://arxiv.org/abs/1209.0873. | ||

[5] | B. C. Carlson, Special Functions of Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. | ||

[6] | D. E. Edmundes, P. Gurka, and J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory 164 (2012), no. 1, 47-56. | ||

[7] | W.-D. Jiang, M.-K. Wang, Y.-M. Chu, Y.-P. Jiang, and F. Qi, Convexity of the generalized sine function and the generalized hyperbolic sine function, J. Approx. Theory 174 (2013), 1-9. | ||

[8] | R. Klén, M. Vuorinen, and X.-H. Zhang, Inequali ties for the generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl. 409 (2014), no. 1, 521-529. | ||

[9] | P. Lindqvist, Some remarkable sine and cosine functions, Ricerche Mat. 44 (1995), no. 2, 269-290 (1996). | ||

[10] | D. S. Mitrinovic, Analytic inequalities, Springer, New York, 1970. | ||

[11] | E. Neumann and J. Sándor, On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities, Math. Inequal. Appl. 13 (2010), no. 4, 715-723. | ||

[12] | F. Qi, D.-W. Niu, and B.-N. Guo, Refinements, generalizations, and applications of Jordan's inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages. | ||

### Article

**New Properties for The Ramanujan’S Continued Fraction of Order 12**

^{1}Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, INDIA

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 90-95

**DOI:**10.12691/tjant-2-3-7

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Chandrashekar Adiga, M. S. Surekha, A. Vanitha. New Properties for The Ramanujan’S Continued Fraction of Order 12.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):90-95. doi: 10.12691/tjant-2-3-7.

Correspondence to: Chandrashekar Adiga, Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, INDIA. Email: c_adiga@hotmail.com

### Abstract

### Keywords

### References

[1] | C. Adiga, B. C. Berndt, S. Bhargava and G. N. Watson, Chapter 16 of Ramanujan’s second notebook: Theta functions and q-series, Mem. Amer. Math. Soc.,315 (1985), 1-91. | ||

[2] | C. Adiga, K. R. Vasuki and N. Bhaskar, Some new modular relations for the cubic functions, South East Asian Bull. Math.,36 (2012), 1-19. | ||

[3] | G. E. Andrews, On q- difference equations for certain well-poised basic hyoergeometric series, Quart. J. Math. (Oxford),19 (1968), 433-447. | ||

[4] | N. D. Baruah and R. Barman, Certain theta function identities and Ramanujan’s modular equations of degree 3, Indian J. Math.,48 (3) (2006), 113-133. | ||

[5] | B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991. | ||

[6] | S. Bhargava, C. Adiga and D. D. Somashekara, Ramanujan’s remarkable summation formula and an interesting convolution identity, Bull. Austral. Math. Soc.,47 (1993), 155-162. | ||

[7] | Boonrod Yuttanan, New properties for the Ramanujan-Göllnitz-Gordon continued fraction, Acta Arithmetric, 151(3) (2012), 293-310. | ||

[8] | H. Göllnitz, Partitionen mit Diffrenzenbedinguggen, J. Reine Angew Math, 225 (1967), 154-190. | ||

[9] | B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J.32 (1965), 741-748. | ||

[10] | M. S. Mahadeva Naika, B. N. Dharmendra and K. Shivashankar, A continued fraction of order twelve, Centr. Eur. J. Math.,6 (3) (2008), 393-404. | ||

[11] | S. Ramanujan, Notebooks (2 volumes), Tata Inst. Fund. Res., Bombay, 1957. | ||

[12] | H. M. Srivastava, Some convolution identities based upon Ramanujan’s bilateral sum, Bull. Austral. Math. Soc.,49 (1994), 433-437. | ||

[13] | K. R. Vasuki, Abdulrawf A. Kahtan, G. Sharth and C. Sathish Kumar, On a continued fraction of order 12, Ukra. Math. J.,62 (12) (2010), 1866-1878. | ||

[14] | K. R. Vasuki, G. Sharth and K. R. Rajanna, Two modular equations for squares of the cubic functions with applications, Note di Math.30 (2) (2010), 61-70. | ||

[15] | K. W. Yang, On the product , J.Austral. Math. Soc., Ser.A 48 (1990), 148-151. | ||

### Article

**On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function**

^{1}Department of Mathematics, Faculty of Science and Arts, Düzce University, Konu-ralp Campus, Düzce-TURKEY

^{2}Department of Mathematics, Faculty of Science, Bartn University, Konuralp Cam-pus, BARTIN-TURKEY

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 85-89

**DOI:**10.12691/tjant-2-3-6

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Mehmet Zeki Sarikaya, Samet Erden. On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):85-89. doi: 10.12691/tjant-2-3-6.

Correspondence to: Mehmet Zeki Sarikaya, Department of Mathematics, Faculty of Science and Arts, Düzce University, Konu-ralp Campus, Düzce-TURKEY. Email: sarikayamz@gmail.com

### Abstract

**In this paper, we extend some estimates of the right hand side of a Hermite- Hadamard-Fejér type inequality for functions whose first derivatives absolute values are convex. The results presented here would provide extensions of those given in earlier works.**

### Keywords

### References

[1] | Z. Dahmani, On Minkowski and Hermite-Hadamard integral inequalities via fractional integration, Ann. Funct. Anal. 1 (1) (2010), 51-58. | ||

[2] | J. Deng and J. Wang, Fractional Hermite-Hadamard inequalities for (α; m)-logarithmically convex functions. | ||

[3] | S. S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett., 11 (5) (1998), 91-95. | ||

[4] | S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. | ||

[5] | S. Hussain, M.A. Latif and M. Alomari, Generalized duble-integral Ostrowski type inequalities on time scales, Appl. Math. Letters, 24 (2011), 1461-1467. | ||

[6] | M. E. Kiris and M. Z. Sarikaya, On the new generalization of Ostrowski type inequality for double integrals, International Journal of Modern Mathematical Sciences, 2014, 9 (3): 221-229. | ||

[7] | L. Fejer, Über die Fourierreihen, II. Math. Naturwiss. Anz Ungar. Akad. Wiss., 24 (1906), 369.390. (Hungarian). | ||

[8] | I. I, scan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional integrals, arXiv preprint arXiv: 1404. 7722 (2014). | ||

[9] | U.S. Krmac, Inequalities for dif ferentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137-146. | ||

[10] | J. Peµcari´c, F. Proschan and Y.L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991. | ||

[11] | M. Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, Hermite -Hadamard.s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (2013) 2403. 2407. | ||

[12] | M. Z. Sarikaya and H. Yildirim, On Hermite-Hadamard type inequalities for Riemann- Liouville fractional integrals, Submited | ||

[13] | M. Z. Sarikaya, On new Hermite Hadamard Fejer Type integral inequalities, Studia Universitatis Babes-Bolyai Mathematica., 57 (2012), No. 3, 377-386. | ||

[14] | K-L. Tseng, G-S. Yang and K-C. Hsu, Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula, Taiwanese J. Math. 15 (4), pp: 1737-1747, 2011. | ||

[15] | C.-L. Wang, X.-H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math. 3 (1982) 567. 570. | ||

[16] | S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., vol. 39, no. 5, pp. 1741. 1749, 2009. | ||

[17] | M. Tunc, On new inequalities for h-convex functions via Riemann-Liouville fractional integration, Filomat 27: 4 (2013), 559. 565. | ||

[18] | J. Wang, X. Li, M. Feckan and Y. Zhou, Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Appl. Anal. (2012). | ||

[19] | B-Y, Xi and F. Qi, Some Hermite-Hadamard type inequalities for differentiable convex func- tions and applications, Hacet. J. Math. Stat. 42 (3), 243. 257 (2013). | ||

[20] | B-Y, Xi and F. Qi, Hermite-Hadamard type inequalities for functions whose derivatives are of convexities, Nonlinear Funct. Anal. Appl. 18 (2), 163. 176 (2013). | ||

[21] | Y. Zhang and J-R. Wang, On some new Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals, Journal of Inequalities and Applications 2013, 2013: 220. | ||

### Article

**On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem**

^{1}Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 80-84

**DOI:**10.12691/tjant-2-3-5

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Mehmet Zeki SARIKAYA, Hüseyin BUDAK. On an Inequality of Ostrowski Type via Variant of Pompeiu's Mean Value Theorem.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):80-84. doi: 10.12691/tjant-2-3-5.

Correspondence to: Mehmet Zeki SARIKAYA, Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey. Email: sarikayamz@gmail.com

### Abstract

### Keywords

### References

[1] | A. M. Acu, A. Babos and F. D. Sofonea, The mean value theorems and inequalities of Ostrowski type. Sci. Stud. Res. Ser. Math. Inform. 21 (2011), no. 1, 5-16. | ||

[2] | S.S. Dragomir, An inequality of Ostrowski type via Pompeiu's mean value theorem, J. of Inequal. in Pure and Appl. Math., 6(3) (2005), Art. 83. | ||

[3] | I. Muntean, Extensions of some mean value theorems, Babes-Bolyai University, Faculty of Mathematics, Research Seminars on Mathematical Analysis, Preprint Nr. 7, 1991, 7-24. | ||

[4] | P.P Pecaric and S. Ungar, On an inequality of Ostrowski type, J. of Inequal. in Pure and Appl. Math., 7(4) (2006), Art. 151. | ||

[5] | E. C. Popa, An inequality of Ostrowski type via a mean value theorem, General Mathematics Vol. 15, No. 1, 2007, 93-100. | ||

[6] | D. Pompeiu, Sur une proposition analogue au théorème des accroissements finis, Mathematica (Cluj, Romania), 22 (1946), 143-146. | ||

[7] | A. Ostrowski, Uber die Absolutabweichung einer differentierbaren Funktionen von ihrem Integralmittelwert, Comment. Math. Helv., 10(1938), 226-227. | ||

[8] | F. Ahmad, N. A. Mir and M.Z. Sarikaya, An inequality of Ostrowski type via variant of Pompeiu's mean value theorem, J. Basic. Appl. Sci. Res., 4(4)204-211, 2014. | ||

### Article

**Using Area Mean Value Theorem to Solve Some Double Integrals**

^{1}Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan

^{2}Department of Information Technology, Nan Jeon University of Science and Technology, Tainan City, Taiwan

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 75-79

**DOI:**10.12691/tjant-2-3-4

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Chii-Huei Yu, Shinn-Der Sheu. Using Area Mean Value Theorem to Solve Some Double Integrals.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):75-79. doi: 10.12691/tjant-2-3-4.

Correspondence to: Chii-Huei Yu, Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan. Email: chiihuei@nju.edu.tw

### Abstract

### Keywords

### References

[1] | A. A. Adams, H. Gottliebsen, S. A. Linton, and U. Martin, “Automated theorem proving in support of computer algebra: symbolic definite integration as a case study,” Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, Canada, pp. 253-260, 1999. | ||

[2] | M. A. Nyblom, “On the evaluation of a definite integral involving nested square root functions,” Rocky Mountain Journal of Mathematics, Vol. 37, No. 4, pp. 1301-1304, 2007. | ||

[3] | C. Oster, “Limit of a definite integral,” SIAM Review, Vol. 33, No. 1, pp. 115-116, 1991. | ||

[4] | C. -H. Yu, “A study of two types of definite integrals with Maple,” Jökull Journal, Vol. 64, No. 2, pp. 543-550, 2014. | ||

[5] | C. -H. Yu, “Evaluating two types of definite integrals using Parseval’s theorem,” Wulfenia Journal, Vol. 21, No. 2, pp. 24-32, 2014. | ||

[6] | C. -H. Yu, “Solving some definite integrals using Parseval’s theorem,” American Journal of Numerical Analysis, Vol. 2, No. 2, pp. 60-64, 2014. | ||

[7] | C. -H. Yu, “Some types of integral problems,” American Journal of Systems and Software, Vol. 2, No. 1, pp. 22-26, 2014. | ||

[8] | C. -H. Yu, “Using Maple to study the double integral problems,” Applied and Computational Mathematics, Vol. 2, No. 2, pp. 28-31, 2013. | ||

[9] | C. -H. Yu, “A study on double Integrals,” International Journal of Research in Information Technology, Vol. 1, Issue. 8, pp. 24-31, 2013. | ||

[10] | C. -H. Yu, “Application of Parseval’s theorem on evaluating some definite integrals,” Turkish Journal of Analysis and Number Theory, Vol. 2, No. 1, pp. 1-5, 2014. | ||

[11] | C. -H. Yu, “Evaluation of two types of integrals using Maple, ”Universal Journal of Applied Science, Vol. 2, No. 2, pp. 39-46, 2014. | ||

[12] | C. -H. Yu, “Studying three types of integrals with Maple, ”American Journal of Computing Research Repository, Vol. 2, No. 1, pp. 19-21, 2014. | ||

[13] | C. -H. Yu, “The application of Parseval’s theorem to integral problems,” Applied Mathematics and Physics, Vol. 2, No. 1, pp. 4-9, 2014. | ||

[14] | C. -H. Yu, “A study of some integral problems using Maple,” Mathematics and Statistics, Vol. 2, No. 1, pp. 1-5, 2014. | ||

[15] | C. -H. Yu, “Solving some definite integrals by using Maple, ”World Journal of Computer Application and Technology, Vol. 2, No. 3, pp. 61-65, 2014. | ||

[16] | C. -H. Yu, “Using Maple to study two types of integrals,” International Journal of Research in Computer Applications and Robotics, Vol. 1, Issue. 4, pp. 14-22, 2013. | ||

[17] | C. -H. Yu, “Solving some integrals with Maple,” International Journal of Research in Aeronautical and Mechanical Engineering, Vol. 1, Issue. 3, pp. 29-35, 2013. | ||

[18] | C. -H. Yu, “A study on integral problems by using Maple,” International Journal of Advanced Research in Computer Science and Software Engineering, Vol. 3, Issue. 7, pp. 41-46, 2013. | ||

[19] | C. -H. Yu, “Evaluating some integrals with Maple,” International Journal of Computer Science and Mobile Computing, Vol. 2, Issue. 7, pp. 66-71, 2013. | ||

[20] | C. -H. Yu, “Application of Maple on evaluation of definite integrals,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 823-827, 2013. | ||

[21] | C. -H. Yu, “Application of Maple on the integral problems,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 849-854, 2013. | ||

[22] | C. -H. Yu, “Using Maple to study the integrals of trigonometric functions,” Proceedings of the 6th IEEE/International Conference on Advanced Infocomm Technology, Taiwan, No. 00294, 2013. | ||

[23] | C. -H. Yu, “A study of the integrals of trigonometric functions with Maple,” Proceedings of the Institute of Industrial Engineers Asian Conference 2013, Taiwan, Springer, Vol. 1, pp. 603-610, 2013. | ||

[24] | C. -H. Yu, “Application of Maple on the integral problem of some type of rational functions,” (in Chinese) Proceedings of the Annual Meeting and Academic Conference for Association of IE, Taiwan, D357-D362, 2012. | ||

[25] | C. -H. Yu, “Application of Maple on some integral problems, ”(in Chinese) Proceedings of the International Conference on Safety & Security Management and Engineering Technology 2012, Taiwan, pp. 290-294, 2012. | ||

[26] | C. -H. Yu, “Application of Maple on some type of integral problem,”(in Chinese) Proceedings of the Ubiquitous-Home Conference 2012, Taiwan, pp.206-210, 2012. | ||

[27] | C. -H. Yu, “Application of Maple on evaluating the closed forms of two types of integrals,”(in Chinese) Proceedings of the 17th Mobile Computing Workshop, Taiwan, ID16, 2012. | ||

[28] | C. -H. Yu, “Application of Maple: taking two special integral problems as examples,”(in Chinese) Proceedings of the 8th International Conference on Knowledge Community, Taiwan, pp.803-811, 2012. | ||

[29] | C. -H. Yu, “Evaluating some types of definite integrals,” American Journal of Software Engineering, Vol. 2, Issue. 1, pp. 13-15, 2014. | ||

[30] | C. -H. Yu and B. -H. Chen, “Solving some types of integrals using Maple,” Universal Journal of Computational Mathematics, Vol. 2, No. 3, pp. 39-47, 2014. | ||

[31] | T. -J. Chen and C. -H. Yu, “A study on the integral problems of trigonometric functions using two methods,” Wulfenia Journal, Vol. 21, No. 4, pp. 76-86, 2014. | ||

[32] | T. -J. Chen and C. -H. Yu, “Fourier series expansions of some definite integrals,” Sylwan Journal, Vol. 158, Issue. 5, pp. 124-131, 2014. | ||

[33] | T. -J. Chen and C. -H. Yu, “Evaluating some definite integrals using generalized Cauchy integral formula,” Mitteilungen Klosterneuburg, Vol. 64, Issue. 5, pp.52-63, 2014. | ||

[34] | R. V. Churchill and J. W. Brown, Complex variables and applications, McGraw-Hill, New York, 1984. | ||

[35] | W. R. Derrick, Introductory complex analysis and applications, Academic Press, New York, 1973. | ||

### Article

**Numerical Computation Based on the Method of Fundamental Solutions for a Cauchy Problem of Heat Equation**

^{1}School of Mathematics and Computer science, Shanxi Normal University, Linfen, China

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 70-74

**DOI:**10.12691/tjant-2-3-3

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Ruihua Cao. Numerical Computation Based on the Method of Fundamental Solutions for a Cauchy Problem of Heat Equation.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):70-74. doi: 10.12691/tjant-2-3-3.

Correspondence to: Ruihua Cao, School of Mathematics and Computer science, Shanxi Normal University, Linfen, China. Email: caoruihua0056@126.com

### Abstract

### Keywords

### References

[1] | L. Ling, T. Takeuchi. Boundary control for inverse Cauchy problems of the Laplace equations. CMES Comput Model Eng Sci, 2008, Vol. 29, No. 1, pp. 45-54. | ||

[2] | M. C. Flemings. Solidification processing. McGraw-Hill, New York, 1974. | ||

[3] | D. A. Porter, K. E. Easteling. Phase Transformations in Metals and Alloys. Chapman Hall, London, 1981. | ||

[4] | D. Lesnic, L Elliott, D. B. Ingham. Application of the boundary element method to inverse heat conduction problems. International Communications in Heat and Mass Transfer, 1996 Vol. 39, No. 7, PP. 1503-1517. | ||

[5] | L Guo, D. Murio. A mollified space-marching finite-difference algorithm for the two-dimensional inverse heat conduction problem with slab symmetry. Inverse problem, 1991, Vol. 7, No. 2, PP. 247-259. | ||

[6] | T. R. Hsu, N. S. Sun, G.G. Chen, Z. L. Gong. Finite element formulation for two-dimensional inverse heat conduction analysis. Journal of Heat Transfer, 1992, Vol. 114. No. 3, PP. 553-557. | ||

[7] | T. Wei, Y. C. Hon. A fundamental solution method for inverse heat conduction problem. Engineering analysis with boundary elements, 2004, Vol. 28, No. 5, pp. 489-495. | ||

[8] | Y. C. Hon, T. Wei. The method of fundamental solution for solving multidimensional inverse heat conduction problems. CMES Compt. Model. Eng. Sci, 2005, Vol. 7, No. 2, pp. 119-132. | ||

[9] | A. N. Tikhonov, V. Y. Arsenin. On the solution of ill-posed problems. John Wiley and Sons, New York, 1977. | ||

[10] | T. Wei, Y. C. Hon, L. Ling. Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Engineering Analysis with Boundary Elements, 2007, Vol. 31, No. 4, pp. 373-385. | ||

[11] | P. C. Hansen. Regularization Tools: a Matlab package for analysis and solution of discrete of ill-posed problems. Numerical Algorithms, 1994, Vol. 6, No. 1-2, PP. 1-35. | ||

### Article

**A Note on Saigo’s Fractional Integral Inequalities**

^{1}School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, People’s Republic of China

^{2}Department of Mathematics, Amity University, Jaipur, India

^{3}Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur, India

^{4}Department of Civil Engineering, College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur, India

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 65-69

**DOI:**10.12691/tjant-2-3-2

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Guotao Wang, Harshvardhan Harsh, S.D. Purohit, Trilok Gupta. A Note on Saigo’s Fractional Integral Inequalities.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):65-69. doi: 10.12691/tjant-2-3-2.

Correspondence to: Guotao Wang, School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi, People’s Republic of China. Email: wgt2512@163.com

### Abstract

### Keywords

### 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. | ||

### Article

**Integral Inequalities of Hermite–Hadamard Type for m-AH Convex Functions**

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

^{2}Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, China

^{3}Institute of Mathematics, Henan Polytechnic University, jiaozuo City, Henan Province, China

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(3), 60-64

**DOI:**10.12691/tjant-2-3-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Tian-Yu Zhang, Feng Qi. Integral Inequalities of Hermite–Hadamard Type for m-AH Convex Functions.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(3):60-64. doi: 10.12691/tjant-2-3-1.

Correspondence to: Tian-Yu Zhang, College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Email: zhangtianyu7010@126.com

### Abstract

### Keywords

### References

[1] | S.S. Dragomir and R.P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 5 (1998), 91-95. | ||

[2] | U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp. 147 (2004), 137-146. | ||

[3] | G. Toader, Some generalizations of the convexity, Proceedings of the Colloquium on Approximation and Optimization, Univ. Cluj-Napoca, Cluj-Napoca, 1985, 329-338. | ||

[4] | S.S. Dragomir, On some new inequalities of Hermite-Hadamard type for m-convex functions, Tamkang J. Math. 33 (2002) 45-55. | ||

[5] | S.S. Dragomir, G. Toader, Some inequalities for m-convex functions, Studia Univ. Babes¸-Bolyai Math. 38 (1993), 21-28. | ||

[6] | M. Klaričič Bakula, M. E. Özdemir, and J. Pečarić, Hadamard type inequalities for m-convex and (a,m)-convex functions, J. Inequal. Pure Appl. Math. 9 (2008), no. 4, Art. 96, 12 pages. | ||

[7] | İ. İşcan, A new generalization of some integral inequalities for (a,m)-convex functions, Mathematical Sciences 7 (2013), 22, 1-8. | ||

[8] | İ. İşcan, New estimates on generalization of some integral inequalities for (a,m)-convex functions, Contemporary Analysis and Applied Mathematics 1 (2013), no. 2, 253-264. | ||

[9] | S.-H. Wang, B.-Y. Xi, and F. Qi, On Hermite-Hadamard type inequalities for (a,m)-convex functions, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 4, 47-56. | ||

[10] | S.-H. Wang, B.-Y. Xi, and F. Qi, Some new inequalities of Hermite-Hadamard type for n-time dierentiable functions which are m-convex, Analysis (Munich) 32 (2012), no. 3, 247-262. | ||

[11] | B.-Y. Xi, R.-F. Bai, and F. Qi, Hermite-Hadamard type inequalities for the m-and (a,m)-geometrically convex functions, Aequationes Math., 84 (2012), no. 3, 261-269. | ||

[12] | V.G. Mihesan. A generalization of the convexity, Seminar on functional equations, approximation and convexity, Cluj-Napoca, 1993. (Romania). | ||