Dear Label_Name,

Greetings! I hope you are well.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I am looking forward to hearing from you.

Kind regards,

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

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

Da-Wei Niu*, Yue-Jin Zhang, Feng Qi

College of Information and Business, Zhongyuan University of Technology, Zhengzhou City, Henan Province, China

Abstract: In the paper, the author present an inequality for bounding the harmonic number in terms of the hyperbolic cosine.

Keywords: Inequality, Euler-Mascheroni constant, Harmonic number, Hyperbolic cosine

Received December 03, 2014; Revised December 08, 2014; Accepted December 14, 2014

Citation: 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, 2 (6). 223-225. January 2014. doi:10.12691/tjant-2-6-6

1. Introduction

The harmonic number is defined as

and it has the following close connections with the Euler-Mascheroni constant :

and

where is the digamma function which is the logarithmic derivative of the classical Euler gamma function

The harmonic number has interesting applications in many areas of mathematics, such as number theory, special functions, and combinatorics. For example, Lagarias proved that the Riemann hypothesis is equivalent to the statement that

for , where denotes the sum of the divisors of n.

In [20], Paule and Schneider obtained the identity

In [2], Alzer presented the inequality

for , where the constants and are the best possible. In [5], Batir gave an inequality

This double inequality was refined in [4] by replacing by 1. It also inspired Mortici to construct a sequence

in [15], which converges to more quickly.

For more information on the harmonic number , please refer to [2,6-19,21-26] and plenty of references therein.

In this paper, we will establish a new double inequality for bounding the harmonic number in terms of the hyperbolic cosine.

Our main result may be stated as the following theorem.

Theorem 1.1. For all positive integers , we have

[picture] | (1.1)

where the constants and are the best impossible.

2. Lemmas

In order to prove Theorem 1.1, we need the following lemmas.

Lemma 2.1 ([[3], p. 384]). Let and be integers, for , we have

where

Lemma 2.2 ([10,22]). For , we have

[picture] | (2.1)

3. Proof of Theorem 1.1

Now we are in a position to prove our Theorem 1.1.

Let

A direct differentiation yields

and

By virtue of inequalities (2.1) and

we acquire

for . This implies that and that is increasing on .

By the asymptotic expansion

as in [[1], p. 259] and the well-known formula

[picture] | (3.1)

we easily find

as . Hence, it follows that

Taking into account that is increasing on reveals

[picture] | (3.2)

Combining (1), (3.1), and (3.2) concludes that the double inequality (1.1) holds for all and that the bounds and in (1.1) are the best impossible. The proof of Theorem 1.1 is complete.

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.
[3]   | H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373-389.
[4]   | H. Alzer, Sharp inequalities for the harmonic numbers, Expo. Math. 24 (2006), no. 4, 385-388.
[5]   | N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 103.
[6]   | C.-P. Chen, Inequalities for the Euler-Mascheroni costant, Appl. Math. Lett. 23 (2010), no. 2, 161-164.
[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.
[10]   | B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103-111.
[11]   | B.-N. Guo and F. Qi, Sharp bounds for harmonic numbers, Appl. Math. Comput. 218 (2011), no. 3, 991-995.
[12]   | B.-N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201-208.
[13]   | E. A. Karatsuba, On the computation of the Euler constant, Numer. Algor. 24 (2000), no. 1-2, 83-97.
[14]   | W.-H. Li, F. Qi, and B.-N. Guo, On proofs for monotonicity of a function involving the psi and exponential functions, Analysis (Munich) 33 (2013), no. 1, 45-50.
[15]   | C. Mortici, A quicker convergence toward the constant with the logarithm term involving the constant e, Carpathian J. Math. 26 (2010), no. 1, 86-91.
[16]   | C. Mortici, Improved convergence towards generalized Euler-Mascheroni constant, Appl. Math. Comput. 215 (2010), no. 9, 3443-3448.
[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.
[23]   | F. Qi and Q.-M. Luo, Complete monotonicity of a function involving the gamma function and applications, Period. Math. Hungar. 69 (2014), no. 2, 159-169.
[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.
[26]   | R. M. Young, Euler's constant, Math. Gaz. 75 (1991), 187-190.