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Turkish Journal of Analysis and Number Theory

## Article

# Hermite-Hadamard Type Inequalities for *s*-Convex Stochastic Processes in the Second Sense

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

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 202-207

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

ERHAN SET, MUHARREM TOMAR, SELAHATTIN MADEN. Hermite-Hadamard Type Inequalities for

*s*-Convex Stochastic Processes in the Second Sense.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):202-207. doi: 10.12691/tjant-2-6-3.

Correspondence to: MUHARREM TOMAR, Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey. Email: muharremtomar@odu.edu.tr

## Abstract

## Keywords

## References

[1] | K. Nikodem, on convex stochastic processes, Aequationes Mathematicae 20 (1980) 184-197. | ||

[2] | A. Skowronski, On some properties of J-convex stochastic processes, Aequationes Mathematicae 44 (1992) 249-258. | ||

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[4] | D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151. | ||

[5] | S.S. Dragomir and S. Fitzpatrick, The Hadamard's inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4) (1999), 687-696. | ||

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[7] | S. Maden, M. Tomar and E. Set, s-convex stochastic processes in the first sense, Pure and Applied Mathematics Letters, in press. | ||

[8] | D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151. | ||

## Article

# Birth of Compound Numbers

^{1,}

^{1}Department of Computer Science, Jamia Hamdard University, Hamdard Nagar, New Delhi, INDIA

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 208-219

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Ranjit Biswas. Birth of Compound Numbers.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):208-219. doi: 10.12691/tjant-2-6-4.

Correspondence to: Ranjit Biswas, Department of Computer Science, Jamia Hamdard University, Hamdard Nagar, New Delhi, INDIA. Email: ranjitbiswas@yahoo.com

## Abstract

## Keywords

## References

[1] | Alperin, J. L., with R. B. Bell, Groups and Representations, Graduate Texts in Mathematics, Vol. 162, Springer-Verlag, New York, 1995. | ||

[2] | Artin, Michael, Algebra, Prentice Hall, New York, 1991. | ||

[3] | Beachy, J. A., and W. D. Blair, Abstract Algebra, 2nd Ed., Waveland Press, Prospect Heights, Ill., 1996. | ||

[4] | Bourbaki, Nicolas, Elements of Mathematics: Algebra I, New York: , 1998. | ||

[5] | Dixon, G.M., Division Algebras: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics, December 2010, Kluwer Academic Publishers, Dordrecht. | ||

[6] | Ellis, G., Rings and Fields, Oxford University Press, 1993. | ||

[7] | Hideyuki Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986 | ||

[8] | Hungerford, T., Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, New York, 1974. | ||

[9] | I. N. Herstein, Topics in Algebra, Wiley Eastern Limited, New Delhi, 2001. | ||

[10] | Jacobson, N., Basic Algebra I, 2nd Ed., W. H. Freeman & Company Publishers, San Francisco, 1985. | ||

[11] | Jacobson, N., Basic Algebra II, 2nd Ed., W. H. Freeman & Company Publishers, San Francisco, 1989. | ||

[12] | Lam, T. Y., Exercises in Classical Ring Theory, Problem Books in Mathematics, Springer-Verlag, New York, 1995. | ||

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[15] | Pierce, Richard S., Associative algebras. Graduate Texts in Mathematics, 88. Studies in the History of Modern Science, 9. Springer-Verlag, New York-Berlin, 1982. | ||

[16] | Ranjit Biswas, Region Algebra, Theory of Objects & Theory of Numbers, International Journal of Algebra, Vol. 6 (28) 2012 page 1371-1417. | ||

[17] | Ranjit Biswas, Calculus Space, International Journal of Algebra, Vol. 7, 2013, No.16, 791-801. | ||

[18] | Van der Waerden, B. L., Algebra, Springer-Verlag, New York, 1991. | ||

[19] | Van der Waerden, Bartel Leendert, Algebra, Berlin, New York: , 1993, ISBN 978-3-540-56799-8. | ||

[20] | Walter Rudin, Real and Complex Analysis, McGraw Hills Education, India, August’ 2006. | ||

## Article

# Note on a Partition Function Which Assumes All Integral Values

^{1,}

^{1}Department of Mathematics, Goa University, Taleigao Plateau, Goa, India

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 220-222

**DOI:**10.12691/tjant-2-6-5

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Manvendra Tamba. Note on a Partition Function Which Assumes All Integral Values.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):220-222. doi: 10.12691/tjant-2-6-5.

Correspondence to: Manvendra Tamba, Department of Mathematics, Goa University, Taleigao Plateau, Goa, India. Email: tamba@unigoa.ac.in

## Abstract

*G*(

*n*) denote the number of partitions of

*n*into distinct parts which are of the form 2

*m*, 3

*m*, 5

*m*, 6

*m*-3, 8

*m*-3, 9

*m*-3 or 11

*m*-3 with parts of the form 2

*m*, 3

*m*, 6

*m*-3, or 11

*m*-3 being even in number minus the number of them with parts of the form 2

*m*, 3

*m*, 6

*m*-3, or 11

*m*-3 being odd in number. In this paper, we prove that

*G*(

*n*) assumes all integral values and does so infinitely often.

## Keywords

## References

[1] | G.E. Andrews, The Theory of Partitions, (G.-C. Rota, Ed.), Encyclopedia of Math. And its Applications, Vol. 2, Addison-Wesley, Reading, MA, 1976. | ||

[2] | G.E. Andrews, Questions and conjectures in partition theory, Amer. Math. Monthly, 93 (1986) 708-711. | ||

[3] | G.E. Andrews, F.J. Dyson and D. Hickerson, Partitions and indefinitequadratic forms, Invent. math., 91 (1988) 391-407. | ||

[4] | J. Lovejoy, Lacunary Partition Functions, Math. Research Letters, 9 (2002) 191-198. | ||

[5] | M.Tamba, On a partition function which assumes all integral values, J.Number Theory 41 (1992) 77-86. | ||

## Article

# A Double Inequality for the Harmonic Number in Terms of the Hyperbolic Cosine

^{1}College of Information and Business, Zhongyuan University of Technology, Zhengzhou City, Henan 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(6), 223-225

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

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Da-Wei Niu, Yue-Jin Zhang, Feng Qi. A Double Inequality for the Harmonic Number in Terms of the Hyperbolic Cosine.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):223-225. doi: 10.12691/tjant-2-6-6.

Correspondence to: Da-Wei Niu, College of Information and Business, Zhongyuan University of Technology, Zhengzhou City, Henan Province, China. Email: nnddww@163.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, 10th printing, Washington, 1972. | ||

[2] | H. Alzer, Inequalities for the harmonic numbers, Math. Z. 267 (2011), no. 1-2, 367-384. | ||

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[7] | C.-P. Chen, Sharpness of Negoi's inequality for the Euler-Mascheroni constant, Bull. Math. Anal. Appl. 3 (2011), no. 1, 134-141. | ||

[8] | C.-P. Chen and C. Mortici, New sequence converging towards the Euler-Mascheroni constant, Comp. Math. Appl. 64 (2012), no. 2, 391-398. | ||

[9] | D. W. DeTemple, A quicker convergence to Euler's constant, Amer. Math. Monthly 100 (1993), no. 5, 468-470. | ||

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[12] | B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208. | ||

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[17] | C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 59 (2010), no. 1, 97-100. | ||

[18] | C. Mortici, On new sequences converging towards the Euler-Mascheroni constant, Comput. Math. Appl. 59 (2010), no. 8, 2610-2614. | ||

[19] | T. Negoi, A faster convergence to Euler's constant, Math. Gaz. 83 (1999), no. 498, 487-489. | ||

[20] | P. Paule and C. Schneider, Computer proofs of a new family of harmonic number identities, Adv. Appl. Math. 31 (2003), no. 2, 359-378. | ||

[21] | F. Qi, Complete monotonicity of functions involving the q-trigamma and q-tetragamma functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 109 (2015), in press. | ||

[22] | F. Qi, R.-Q. Cui, C.-P. Chen and B.-N. Guo, Some completely monotonic functions involving polygamma functions and an application, J. Math. Anal. Appl. 310 (2005), no. 1, 303-308. | ||

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[24] | A. Sîntǎmǎrian A generalization of Euler's constant, Numer. Algor. 46 (2007) no. 2, 141-151. | ||

[25] | M. B. Villarino, Ramanujan's harmonic number expansion into negative powers of a triangular num- ber, J. Inequal. Pure Appl. Math. 9 (2008), no. 3, Art. 89. | ||

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## Article

# Generalized Inequalities Related to the Classical Euler’s Gamma Function

^{1,}

^{1}Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 226-229

**DOI:**10.12691/tjant-2-6-7

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Kwara Nantomah. Generalized Inequalities Related to the Classical Euler’s Gamma Function.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):226-229. doi: 10.12691/tjant-2-6-7.

Correspondence to: Kwara Nantomah, Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana. Email: mykwarasoft@yahoo.com, knantomah@uds.edu.gh

## Abstract

## Keywords

## References

[1] | R. Diaz and E. Pariguan, On hypergeometric functions and Pachhammer k-symbol, Divulgaciones Matematicas 15(2)(2007), 179-192. | ||

[2] | T. Mansour, Some inequalities for the q-Gamma Function, J. Ineq. Pure Appl. Math. 9(1)(2008), Art. 18. | ||

[3] | F. Merovci, Power Product Inequalities for the Γ_{k} Function, Int. Journal of Math. Analysis, 4(21)(2010), 1007-1012. | ||

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[5] | R. Diaz and C. Teruel, q,k-generalized gamma and beta functions, J. Nonlin. Math. Phys. 12(2005), 118-134. | ||

[6] | V. Krasniqi, T. Mansour and A. Sh. Shabani, Some Monotonicity Properties and Inequalities for Γ and ζ Functions, Mathematical Communications 15(2)(2010), 365-376. | ||

[7] | K. Nantomah, On Certain Inequalities Concerning the Classical Euler's Gamma Function, Advances in Inequalities and Applications, Vol. 2014 (2014) Art ID 42. | ||

[8] | K. Nantomah and M. M. Iddrisu, Some Inequalities Involving the Ratio of Gamma Functions, Int. Journal of Math. Analysis 8(12)(2014), 555-560. | ||

[9] | K. Nantomah, M. M. Iddrisu and E. Prempeh, Generalization of Some Inequalities for theRatio of Gamma Functions, Int. Journal of Math. Analysis, 8(18)(2014), 895-900. | ||

[10] | K. Nantomah and E. Prempeh, Generalizations of Some Inequalities for the p-Gamma, q-Gamma and k-Gamma Functions, Electron. J. Math. Anal. Appl. 3(1)(2015),158-163. | ||

[11] | K. Nantomah and E. Prempeh, Some Sharp Inequalities for the Ratio of Gamma Functions, Math. Aeterna, 4(5)(2014), 501-507. | ||

[12] | K. Nantomah and E. Prempeh, Generalizations of Some Sharp Inequalities for the Ratio of Gamma Functions, Math. Aeterna, 4(5)(2014), 539-544. | ||

## Article

# On the Error Term for the Number of Integral Ideals in Galois Extensions

^{1,}

^{1}School of Mathematics, Hefei University of Technology, Hefei, China

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 230-232

**DOI:**10.12691/tjant-2-6-8

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Sanying Shi. On the Error Term for the Number of Integral Ideals in Galois Extensions.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):230-232. doi: 10.12691/tjant-2-6-8.

Correspondence to: Sanying Shi, School of Mathematics, Hefei University of Technology, Hefei, China. Email: vera123_99@hotmail.com

## Abstract

## Keywords

## References

[1] | Chandraseknaran K., Good A., On the number of integral ideals in Galois extensions, Monatsh. Math., 95. 99-109. 1983. | ||

[2] | Heath-Brown D. R., The number of Abelian groups of order at most x, Journtes Arithmttiques, Luminy 1989. | ||

[3] | Huxley M. N., Watt N., The number of ideals in a quadratic field II, Israel J. Math. Part A, 120, 125-153. 2000, | ||

[4] | Ivic A., The number of finite non-isomorphic Abelian groups in mean square, Hardy-Ramanujan J., 9, 17-23. 1986. | ||

[5] | Ivic A., On the Error Term for the Counting Functions of Finite Abelian Groups, Monatsh. Math. 114, 115-124. 1992. | ||

[6] | Iwaniec H., Kowalski E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, 204-216. 2004. | ||

[7] | Landau E., Einführung in die elementare and analytische Theorie der algebraischen Zahlen und der Ideale, Teubner, 1927. | ||

[8] | Lü G., Wang Y., Note on the number of integral ideals in Galois extension, Sci. China Ser. A, 53, 2417-2424. 2010. | ||

[9] | Müller W., On the distribution of ideals in cubic number fields, Monatsh. Math., 106, 211-219. 1988. | ||

[10] | Nowak W.G., On the distribution of integral ideals in algebraic number theory fields, Math. Nachr., 161, 59-74. 1993. | ||

## Article

# Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

^{1}Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India

^{2}Department of Mathematical Sciences and Computer application, Bundelkhand University, Jhansi (U. P.)

^{3}Department of Mathematics, Mandsaur Institute of Technology, Mandsaur (M. P.) India

*Turkish Journal of Analysis and Number Theory*.

**2014**, 2(6), 233-238

**DOI:**10.12691/tjant-2-6-9

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Yogesh Kumar Gupta, Mamta Singh, Omprakash Sikhwal. Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences.

*Turkish Journal of Analysis and Number Theory*. 2014; 2(6):233-238. doi: 10.12691/tjant-2-6-9.

Correspondence to: Yogesh Kumar Gupta, Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India. Email: yogeshgupta.880@rediffmail.com

## Abstract

_{n}

_{=}F

_{n}

_{-1}+F

_{n}

_{-2, }, and F

_{0}=0, F

_{1}=1, where F

_{n}is a n

^{th }number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation B

_{n}

_{=}B

_{n}

_{-1}+B

_{n}

_{-2,}with B

_{0}=2s, B

_{1}=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

## Keywords

## References

[1] | A. F. Horadam: A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68. (5), 1961, 455-459. | ||

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[3] | A.T. Benjamin and D. Walton, Counting on Chebyshev polynomials, Math. Mag. 82, 2009, 117-126. | ||

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[5] | B. Singh, Omprakash Sikhwal, and Yogesh Kumar Gupta, “Generalized Fibonacci-Lucas Sequence, Turkish Journal of Analysis and Number Theory, Vol.2, No.6. (2014), 193-197. | ||

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[10] | M. Singh, Y. Gupta, O. Sikhwal, Generalized Fibonacci-Lucas Sequences its Properties, Global Journal of Mathematical Analysis, 2 (3) 2014, 160-168. | ||

[11] | M. Singh, Y. Gupta, O. Sikhwal, “Identities of Generalized Fibonacci-Like Sequence.” Turkish Journal of Analysis and Number Theory, vol.2, no. 5 (2014): 170-175. doi:10.12691/tjant 2-5-3. | ||

[12] | S. Falcon and A. Plaza: On the Fibonacci K- Numbers, Chaos, Solutions & Fractals, Vol. 32 (5), 2007, 1615-1624. | ||

[13] | Singh, M., Sikhwal, O., and Gupta, Y., Generalized Fibonacci-Lucas Polynomials, International Journal of Advanced Mathematical Sciences, 2 (1) (2014), 81-87 | ||

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[16] | Y. Gupta, M. Singh, and O. Sikhwal, Generalized Fibonacci-Like Polynomials and Some Identities Global Journal of Mathematical Analysis, 2 (4) (2014) 249-258. | ||

## Article

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

^{1}School 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

## Keywords

## References

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[21] | H.Silverman, Univalent functions with negative coefficients, Proc.Amer.Math.Soc., 51 (1975), 109-116. | ||

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## Article

# Common Fixed Point Results for Generalized Symmetric Meir-Keeler Contraction

^{1}Department 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

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## References

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## Article

# On the Numerical Regularity in the aspect of Prime Numbers

^{1,}

^{1}Mathematical 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

## 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. | ||

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