**Turkish Journal of Analysis and Number Theory:**Latest Articles More >>

## Article

# Determinantal Identities of Fibonacci, Fibonacci Like and Lucas Numbers

^{1}Government College, Kannod(M.P.), India

^{2}School of studies in Mathematics

^{3}Vikram University Ujjain, India

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 110-112

**DOI:**10.12691/tjant-2-4-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Sanjay Harne, V.H. Badshah, Sapna Sethiya. Determinantal Identities of Fibonacci, Fibonacci Like and Lucas Numbers.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):110-112. doi: 10.12691/tjant-2-4-1.

Correspondence to: Sapna Sethiya, Vikram University Ujjain, India. Email: sapna.sethiya11@gmail.com

## Abstract

## Keywords

## References

[1] | Benjamin A., Cameron N. and Quinn J.: Fibonacci Determinants- A Combinatorial Approach, Fibonacci Quarterly, 39-55 (1), 2007, Vol 45. | ||

[2] | Bicknell-Johnson M. and Spears C. P.:Classes Of Identities For the Generalized Fibonacci number G_{n}= G_{n-1 }+ G_{n-2}, n ≥ 2. from Matrices with Constantvalued Determinants, Fibonacci Quarterly, 121-128 (2), 1996, Vol. 34. | ||

[3] | B. Singh, O. Sikhwal, and S. Bhatnagar, Fibonacci-Like Sequence and its properties, Int.J. Contemp Math.Sciences, Vol.5, 2010, No.18, 857-868. | ||

[4] | Cahill N. and Narayan D.:Fibonacci and Lucas numbers s Tridigonal Matrix Determinants, Fibonacci Quarterly, 216-221 (3), 2004, Vol. 42. | ||

[5] | Koshy T.:Fibonacci and Lucas Numbers with Applications, Wiley, 2001. | ||

[6] | Krattenthaler C.: Advanced determinant calculus, Seminaire Lotharingien Combin, Article, b42q, 67, 1999. | ||

[7] | Krattenthaler C.: Advanced determinant calculus: A Complement, Liner Algebra Appl., 68-166. | ||

[8] | Macfarlane A. J.: Use of Determinants to Present Identities Involving Fiboncci and Related Numbers, Fibonacci Quarterly, 68-7648(1), 2010, Vol. 48. | ||

[9] | Spivey M. Z.: Fibonacci Identities via the Determinant sum property, College Mathematics Journal, 286-289 (4), 2006, Vol. 37. | ||

## Article

# On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are S-Geometrically Convex

^{1}Mustafa Kemal University, Faculty of Science and Arts, Department of Mathematics, Hatay, Turkey

^{2}The Institute for Graduate Studies in Sciences and Engineering, Kilis 7 Aralk University, Kilis, Turkey

^{3}AGRI Ibrahim Çeçen University, Faculty of Science and Arts, Department of Mathematics, AGRI, Turkey

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 113-118

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

MEVLÜT TUNÇ, IBRAHİM KARABAYIR, EBRU YÜKSEL. On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are S-Geometrically Convex.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):113-118. doi: 10.12691/tjant-2-4-2.

Correspondence to: MEVLÜT TUNÇ, Mustafa Kemal University, Faculty of Science and Arts, Department of Mathematics, Hatay, Turkey. Email: mevluttttunc@gmail.com

## Abstract

## Keywords

## References

[1] | M. Alomari, M. Darus, S. S. Dragomir: New inequalities of Hermite-Hadamard type for func-tions whose second derivatives absolute values are quasi-convex, Tamkang J. Math., Vol. 41 No. 4 (2010/12), 353-359. | ||

[2] | S.S. Dragomir, R.P. Agarwal: Two inequalities for differentiable mappings and applicationsto special means of real numbers and to trapezoidal formula. Appl Math Lett, Vol. 11 No: 5, (1998) 91.95. | ||

[3] | Hermite-Hadamard in-equalities and applications, RGMIA monographs, Victoria University, 2000. [Online: http://www.staff.vu.edu.au/RGMIA/monographs/hermite-hadamard.html]. | ||

[4] | J. Hadamard: Étude sur les propriétés des fonctions entières et en particulier d.une function considerée par Riemann, J. Math Pures Appl., 58, (1893) 171. 215. | ||

[5] | H. Hudzik and L. Maligranda: Some remarks on s-convex functions, Aequationes Math., Vol. 48 (1994), 100-111. | ||

[6] | İ. İscan, Some New Hermite-Hadamard Type Inequalities for Geometrically Convex Functions, Mathematics and Statistics, 1 (2): 86-91, 2013. | ||

[7] | İ. İscan, On Some New Hermite-Hadamard type inequalities for s-geometrically convex functions, International Journal of Mathematics and Mathematical Sciences, Volume 2014 (2014), Article ID 163901, 8 pages. | ||

[8] | D. S. Mitrinovi´c, J. Peµcari´c and A. M. Fink: Classical and new inequalities in analysis, Kluwer Academic, Dordrecht, 1993. | ||

[9] | J. E. Peµcari´c, F. Proschan and Y. L. Tong: Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992. | ||

[10] | M. Tunç: On some new inequalities for convex fonctions, Turk. J. Math. 36 (2012), 245-251. | ||

[11] | B.-Y. Xi, R.-F. Bai and F. Qi: Hermite-Hadamard type inequalities for the m- and (α;m)-geometrically convex functions. Aequationes Math. | ||

[12] | T.-Y. Zhang, A.-P. Ji and F. Qi: On Integral nequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Abstract and Applied Analysis. | ||

[13] | T.-Y. Zhang, M. Tunç, A.-P. Ji, B.-Y. Xi: Erratum to. On Integral Inequalities of Hermite-Hadamard Type for s-Geometrically Convex Functions. Abstract and Applied Analysis. Volume 2014, Article ID 294739, 5 pages. | ||

## Article

# Some Properties of *k*-Jacobsthal Numbers with Arithmetic Indexes

^{1}School of Studies in Mathematics, Vikram University, Ujjain (India)

^{2}College of Horticulture, Mandsaur (M.P.)

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 119-124

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Deepika Jhala, G.P.S. Rathore, Kiran Sisodiya. Some Properties of

*k*-Jacobsthal Numbers with Arithmetic Indexes.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):119-124. doi: 10.12691/tjant-2-4-3.

Correspondence to: Deepika Jhala, School of Studies in Mathematics, Vikram University, Ujjain (India). Email: jhala.deepika28@gmail.com

## Abstract

## Keywords

## References

[1] | Bolat C., Kose H., On the Properties of k-Fibonacci Numbers. Int. J.Contemp. Math. Sciences 2010, 22 (5), 1097-1105. | ||

[2] | Campos H., Catarino P., Aires A.P., Vasco P. and Borges A., On Some Identities of k-Jacobsthal-Lucas Numbers, Int. Journal of Math. Analysis, 2014, 8 (10), 489-494. | ||

[3] | Catarino P., On Some Identities and Generating Functions for k-Pell Numbers. Int. Journal of Math. Analysis 2013, 7 (38), 1877-1884. | ||

[4] | Catarino P., Vasco P., Modified k-Pell Sequence: Some Identities and Ordinary Generating Function. Applied Mathematical Sciences 2013, 7 (121), 6031-6037. | ||

[5] | Catarino P., Vasco P., On some Identities and Generating Functions for k-Pell-Lucas Sequence. Applied Mathematical Sciences 2013, 7 (98), 4867-4873. | ||

[6] | Catarino P., On Some Identities for k-Fibonacci Sequence. International Journal of Contemporary mathematical Sciences 2014, 9 (1), 37-42. | ||

[7] | Falcon S. and Plaza A., On k-Fibonacci Numbers of Arithmetic Indexes, Applied Mathematics and Computation, 2009, 208, 180-185. | ||

[8] | Falcon S., On the k-Lucas Numbers, International Journal of Contemporary mathematical Sciences 2011, 6 (21), 1039-1050. | ||

[9] | Falcon S., On the k-Lucas Numbers of Arithmetic Indexes, Applied Mathematics, 2012, 3, 1202-1206. | ||

[10] | Horadam A. F., A Generalized Fibonacci Sequence, The American Mathematical Monthly, 1961, 68, 455-459. | ||

[11] | Horadam A.F., Jacobsthal Representation Numbers. Fibonacci Quarterly, 1996, 34 (1), 40-54. | ||

[12] | Jaiswal D. V., On a Generalized Fibonacci Sequence, Labdev Journal of Science and Technology, Part A, 1969, 7, 67-71. | ||

[13] | Jhala D., Sisodiya K., Rathore G.P.S., On Some Identities for k-Jacobsthal Numbers. Int. Journal of Math. Analysis 2013, 7 (2), 551-556. | ||

[14] | Lee G. Y., Lee S. G., Kim J. S., Shin H. K., The Binet Formula and the Representation of k-Generalized Fibonacci Numbers, Fibonacci Quarterly, 2001, 39 (2), 158-164. | ||

[15] | Lee G. Y., Lee S. G., Shin H. K., On the k-Ggeneralized Fibonacci Matrix Q_{k}, Linear Algebra Applications, 1997, 251, 73-88. | ||

[16] | Slone NJA, The On-Line Encyclopedia of Integer Sequences, (2006), www.research.att.com/~njas/sequences/. | ||

[17] | Walton J. E. and Horadam A. F., Some Further Identities for the Generalized Fibonacci sequence, Fibonacci Quarterly, 1974, 12 (3), 272-280. | ||

[18] | Zai L. J. and Sheng L. J., Some Properties of the Generalization of the Fibonacci Sequence, Fibonacci Quarterly, 1987, 25 (2), 111-117. | ||

## Article

# Generating Function for M(m,n)

^{1}Senior Lecturer, Department of Mathematics, Raozan University College, Bangladesh

^{2}Premier University, Chittagong, Bangladesh

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 125-129

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Sabuj Das, Haradhan Kumar Mohajan. Generating Function for M(m,n).

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):125-129. doi: 10.12691/tjant-2-4-4.

Correspondence to: Haradhan Kumar Mohajan, Premier University, Chittagong, Bangladesh. Email: haradhan_km@yahoo.com

## Abstract

*x*in the right hand side of the equation is an algebraic relation in terms of

*z*. The exponent of

*z*represents the crank of partitions of a positive integral value of

*n*and also shows that the sum of weights of corresponding partitions of

*n*is the sum of ordinary partitions of

*n*and it is equal to the number of partitions of

*n*with crank

*m*. This paper shows how to prove the Theorem “The number of partitions π of

*n*with crank C(π)=m is M(m,n) for all n>1.”

## Keywords

## References

[1] | Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. 2 (G-c, Rotaed) Addison-Wesley, Reading, mass, 1976 (Reissued, Cambridge University, Press, London and New York 1985). 1985. | ||

[2] | Andrews, G.E. and Garvan, F.G., Dyson’s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 18(2): 167-171. 1988. | ||

[3] | Atkin, A.O.L. and Swinnerton-Dyer, P., Some Properties of Partitions, Proc. London Math. Soc. 3(4): 84-106. 1954. | ||

[4] | Garvan, F.G., Ramanujan Revisited, Proceeding of the Centenary Conference, University of Illinois, Urban-Champion. 1988. | ||

[5] | Garvan, F.G.. Dyson’s Rank Function and Andrews’ spt-function, University of Florida, Seminar Paper Presented in the University of Newcastle on 20 August 2013. 2013. | ||

## Article

# On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application

^{1}Department of Mathematics, Seth Motilal (P.G.) College, Jhunjhunu, Rajasthan, India

^{2}Department of Mathematics,Government College, Kaladera, Jaipur, Rajasthan , India

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 130-133

**DOI:**10.12691/tjant-2-4-5

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Harmendra Kumar Mandia, Yashwant Singh. On an Integral Involving Bessel Polynomials and -Function of Two Variables and Its Application.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):130-133. doi: 10.12691/tjant-2-4-5.

Correspondence to: Yashwant Singh, Department of Mathematics,Government College, Kaladera, Jaipur, Rajasthan , India. Email: mandiaharmendra@gmail.com; dryashu23@yahoo.in

## Abstract

## Keywords

## References

[1] | Bajpai, S.D. and Al-Hawaj, A.Y.; Application of Bessel polynomials involving generalized hypergeometric functions, J.Indian Acad. Math., vol.13 (1), (1991), 1-5. | ||

[2] | Erdelyi, A. et. al.; Higher Transcendental Functions, vol.1, McGraw-Hill, New York, 1953. | ||

[3] | Erdelyi, A. et. al.; Tables of Integral Transforms, vol.2, McGraw-Hill, New York, 1954. | ||

[4] | Exton, H.; Handbook of Hypergeometric Integrals, ELLIS Harwood Ltd., Chichester, 1978. | ||

[5] | Inayat-Hussain, A.A.; New properties of hypergeometric series derivable from Feynman integrals: II A generalization of the H-function, J. Phys. A. Math. Gen. 20 (1987). | ||

[6] | Mathai, A.M. and Saxena, R.K.; Lecture Notes in Maths. 348, Generalized Hypergeometric Functions With Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin, 1973. | ||

[7] | Mittal, P.K. and Gupta, K.C.; An integral involving generalized function of two variables. Proc. Indian Acad. Sci. Sect. A( 75), (1961), 67-73. | ||

[8] | Singh,Y. and Mandia, H. ; A study of -function of two variables, International Journal of Innovative research in science, engineering and technology,Vol.2,(9),(2013), 4914-4921. | ||

## Article

# Refinements and Sharpening of some Huygens and Wilker Type Inequalities

^{1}Department of Information Engineering, Weihai Vocational University, Weihai, Shandong, China

^{2}Department of Mathematics, Chongqing Normal University, Chongqing City, China

^{3}College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China

^{4}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(4), 134-139

**DOI:**10.12691/tjant-2-4-6

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Wei-Dong Jiang, Qiu-Ming Luo, Feng Qi. Refinements and Sharpening of some Huygens and Wilker Type Inequalities.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):134-139. doi: 10.12691/tjant-2-4-6.

Correspondence to: Feng Qi, College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, China. Email: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com

## Abstract

## Keywords

## References

[1] | M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 4th printing, with corrections, Washington, 1965. | ||

[2] | G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. | ||

[3] | H. Alzer and S.-L. Qiu, Monotonicity theorems and inequalities for complete elliptic integrals, J. Comput. Appl. Math. 172 (2004), no. 2, 289-312. | ||

[4] | B.-N. Guo, W. Li, and F. Qi, Proofs of Wilker's inequalities involving trigonometric functions, The 7th International Conference on Nonlinear Functional Analysis and Applications, Chinju, South Korea, August 6-10, 2001; Inequality Theory and Applications, Volume 3, Yeol Je Cho, Jong Kyu Kim, and Sever S. Dragomir (Eds), Nova Science Publishers, Hauppauge, NY, pp. 109-112. | ||

[5] | B.-N. Guo, B.-M. Qiao, F. Qi, and W. Li, On newproofs of Wilker's inequalities involving trigonometric functions, Math. Inequal. Appl. 6 (2003), no. 1, 19-22. | ||

[6] | Z.-H. Huo, D.-W. Niu, J. Cao, and F. Qi, A generalization of Jordan's inequality and an application, Hacet. J. Math. Stat. 40 (2011), no. 1, 53-61. | ||

[7] | W.-D. Jiang, Q.-M. Luo, and F. Qi. Refinements and sharpening of some Huygens and Wilker type inequalities, available online at http://arxiv.org/abs/1201.6477. | ||

[8] | C. Mortici, The natural approach of Wilker-Cusa-Huygens inequalities, Math. Inequal. Appl. 14 (2011), no. 3, 535-541. | ||

[9] | E. Neuman, On Wilker and Huygnes type inequalities, Math. Inequal. Appl. 15 (2012), no. 2, 271-279. | ||

[10] | E. Neuman and J. S_andor, 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. | ||

[11] | D.-W. Niu, J. Cao, and F. Qi, Generalizations of Jordan's inequality and concerned relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 85-98. | ||

[12] | D.-W. Niu, Z.-H. Huo, J. Cao, and F. Qi, A general refinement of Jordan's inequality and a refinement of L. Yang's inequality, Integral Transforms Spec. Funct. 19 (2008), no. 3, 157-164. | ||

[13] | F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff's integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517-528. | ||

[14] | 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. | ||

[15] | F. Qi and A. Sofo, An alternative and united proof of a double inequality for bounding the arithmeticgeometric mean, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 71 (2009), no. 3, 69-76. | ||

[16] | J. S_andor and M. Bencze, On Huygens' trigonometric inequality, RGMIA Res. Rep. Coll. 8 (2005), no. 3, Art. 14; Available online at http://rgmia. org/v8n3.php. | ||

[17] | J. S. Sumner, A. A. Jagers, M. Vowe, and J. Anglesio, Inequalities involving trigonometric functions, Amer. Math. Monthly 98 (1991), no. 3, 264-267. | ||

[18] | J. B. Wilker, Problem E 3306, Amer. Math. Monthly 96 (1989), no. 1, 55. | ||

[19] | S.-H. Wu and H. M. Srivastava, A further refinement of Wilker's inequality, Integral Transforms Spec. Funct. 19 (2008), no. 10, 757-765. | ||

[20] | L. Zhu, Some new Wilker-type inequalities for circular and hyperbolic functions, Abstr. Appl. Anal. 2009 (2009), Article ID 485842, 9 pages. | ||

## Article

# Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex

^{1}Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

^{2}College of Science, Department of Mathematics, University of Hail, Hail, Saudi Arabia

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

^{4}Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 140-146

**DOI:**10.12691/tjant-2-4-7

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Feng Qi, Muhammad Amer Latif, Wen-Hui Li, Sabir Hussain. Some Integral Inequalities of Hermite-Hadamard Type for Functions Whose n-times Derivatives are (α,m)-Convex.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):140-146. doi: 10.12691/tjant-2-4-7.

Correspondence to: Feng Qi, Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China. Email: qifeng618@gmail.com,

## Abstract

## Keywords

## References

[1] | R.-F. Bai, F. Qi, and B.-Y. Xi, Hermite-Hadamard type inequalities for the m- and (α,m)-logarithmically convex functions, Filomat 27 (2013), no. 1, 1-7. | ||

[2] | S.-P. Bai, S.-H. Wang, and F. Qi, Some Hermite-Hadamard type inequalities for n-time differentiable (α,m)-convex functions, J. Inequal. Appl. 2012, 2012:267, 11 pages. | ||

[3] | L. Chun and F. Qi, Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, J. Inequal. Appl. 2013, 2013:451, 10 pages. | ||

[4] | 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 | ||

[5] | J. Hua, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for geometric-arithmetically s-convex functions, Commun. Korean Math. Soc. 29 (2014), no. 1, 51-63. | ||

[6] | J. Hua, B.-Y. Xi, and F. Qi, Some new inequalities of Simpson type for strongly s-convex functions, Afrika Mat. (2014), in press. | ||

[7] | D.-Y. Hwang, Some inequalities for n-time differentiable mappings and applications, Kyugpook Math. J. 43 (2003), no. 3, 335-343. | ||

[8] | M. A. Latif and S. Hussain, New inequalities of Hermite-Hadamard type for n-time differentiable (α,m)-convex functions with applications to special means, RGMIA Res. Rep. Coll. 16 (2013), Art. 17, 12 pages; Available online at http://rgmia.org/v16.php | ||

[9] | W.-H. Li and F. Qi, Hermite-Hadamard type in-equalities of functions whose derivatives of n-th or-der are (α,m)-convex. | ||

[10] | W.-H. Li and F. Qi, Some Hermite-Hadamard type inequalities for functions whose n-th derivatives are (α,m)-convex, Filomat 27 (2013), no. 8, 1575-1582. | ||

[11] | V. G. Mihe_san, A generalization of the convexity, Seminar on Functional Equations, Approx. Convex, Cluj-Napoca, 1993. (Romania) | ||

[12] | C. P. Niculescu and L.-E. Persson, Convex Functions and their Applications, CMS Books in Mathematics, Springer-Verlag, 2005. | ||

[13] | M. E. Ozdemir, M. Avci, and H. Kavurmaci, Hermite-Hadamard-type ineuqlities via (α,m)- convexity, Comput. Math. Appl. 61 (2011), no. 9, 2614-2620. | ||

[14] | M. E. Ozdemir, M. Avci, and E. Set, On some inequalities of Hermite-Hadamard type via m-convexity, Appl. Math. Lett. 23 (2010), no. 9, 1065-1070. | ||

[15] | 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. | ||

[16] | F. Qi and B.-Y. Xi, Some integral inequalities of Simpson type for GA-"-convex functions, Georgian Math. J. 20 (2013), no. 4, 775-78. | ||

[17] | M. Z. Sarikaya and N. Aktan, On the generalization of some integral inequalities and their applications. | ||

[18] | D.-P. Shi, B.-Y. Xi, and F. Qi, Hermite-Hadamard type inequalities for (m,h1,h2)-convex functions via Riemann-Liouville fractional integrals, Turkish J. Anal. Number Theory 2 (2014), no. 1, 22-27. | ||

[19] | Y. Shuang, Y. Wang, and F. Qi, Some inequalities of Hermite-Hadamard type for functions whose third derivatives are (α;m)-convex, J. Comput. Anal. Appl. 17 (2014), no. 2, 272-279. | ||

[20] | G. Toader, Some generalizations of the convexity, Univ. Cluj-Napoca, Cluj-Napoc. 1985, 329-338. | ||

[21] | S.-H. Wang and F. Qi, Hermite-Hadamard type in-equalities for n-times differentiable and preinvex functions, J. Inequal. Appl. 2014, 2014: 49, 9 pages. | ||

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

[24] | Y. Wang, B.-Y. Xi, and F. Qi, Hermite-Hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex, Matematiche (Catania) 69 (2014), no. 1, 89-96. | ||

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[27] | B.-Y. Xi, J. Hua, and F. Qi, Hermite-Hadamard type inequalities for extended s-convex functions on the co-ordinates in a rectangle, J. Appl. Anal. 20 (2014), no. 1, 29-39. | ||

[28] | B.-Y. Xi and F. Qi, Some Hermite-Hadamard type inequalities for geometrically quasi-convex functions, Proc. Indian Acad. Sci. Math. Sci. (2014), in press. | ||

[29] | B.-Y. Xi and F. Qi, Some inequalities of Hermite-Hadamard type for h-convex functions, Adv. Inequal. Appl. 2 (2013), no. 1, 1-15. | ||

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[31] | B.-Y. Xi, S.-H. Wang, and F. Qi, Properties and inequalities for the h- and (h; m)-logarithmically convex functions, Creat. Math. Inform. 23 (2014), no. 1, 123-130. | ||

[32] | B.-Y. Xi, S.-H. Wang, and F. Qi, Some inequalities for (h,m)-convex functions, J. Inequal. Appl. 2014, 2014: 100, 12 pages. | ||

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## Article

# A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

^{2}Department of Mathematics, Faculty of Science, Mansoura University, ansoura 35516, Egypt

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(4), 147-151

**DOI:**10.12691/tjant-2-4-8

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Mansour Mahmoud. A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(4):147-151. doi: 10.12691/tjant-2-4-8.

Correspondence to: Mansour Mahmoud, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Email: mansour@mans.edu.eg

## Abstract

## Keywords

## References

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## Article

# Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity

^{1}Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(5), 152-164

**DOI:**10.12691/tjant-2-5-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Feng Qi. Bounds for the Ratio of Two Gamma Functions: from Gautschi’s and Kershaw’s Inequalities to Complete Monotonicity.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(5):152-164. doi: 10.12691/tjant-2-5-1.

Correspondence to: Feng Qi, Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, China. Email: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com

## Abstract

*q-*gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.

## Keywords

## References

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## Article

# On the Simpson’s Inequality for Convex Functions on the Co-Ordinates

^{1}Ataturk University, K.K. Education Faculty, Department of Mathematics, Erzurum, Turkey

^{2}Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey

^{3}Yüzüncü Yil University, Education Faculty, Department of Mathematics, Van, Turkey

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(5), 165-169

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

M. EMIN ÖZDEMIR, AHMET OCAK AKDEMIR, HAVVA KAVURMACI. On the Simpson’s Inequality for Convex Functions on the Co-Ordinates.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(5):165-169. doi: 10.12691/tjant-2-5-2.

Correspondence to: AHMET OCAK AKDEMIR, Ağri İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, AĞRI, Turkey. Email: aocakakdemir@gmail.com

## Abstract

## Keywords

## References

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