Keywords: Inequality, EulerMascheroni constant, Harmonic number, Hyperbolic cosine
Turkish Journal of Analysis and Number Theory, 2014 2 (6),
pp 223225.
DOI: 10.12691/tjant266
Received November 03, 2014; Revised December 03, 2014; Accepted December 11, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
The harmonic number is defined as
and it has the following close connections with the EulerMascheroni 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 [2,619,2126] 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
 (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
 (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 wellknown formula
 (3.1) 
we easily find
as . Hence, it follows that
Taking into account that is increasing on reveals
 (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. 
 In article  

[2]  H. Alzer, Inequalities for the harmonic numbers, Math. Z. 267 (2011), no. 12, 367384. 
 In article  CrossRef 

[3]  H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (1997), no. 217, 373389. 
 In article  CrossRef 

[4]  H. Alzer, Sharp inequalities for the harmonic numbers, Expo. Math. 24 (2006), no. 4, 385388. 
 In article  CrossRef 

[5]  N. Batir, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 103. 
 In article  

[6]  C.P. Chen, Inequalities for the EulerMascheroni costant, Appl. Math. Lett. 23 (2010), no. 2, 161164. 
 In article  CrossRef 

[7]  C.P. Chen, Sharpness of Negoi's inequality for the EulerMascheroni constant, Bull. Math. Anal. Appl. 3 (2011), no. 1, 134141. 
 In article  

[8]  C.P. Chen and C. Mortici, New sequence converging towards the EulerMascheroni constant, Comp. Math. Appl. 64 (2012), no. 2, 391398. 
 In article  CrossRef 

[9]  D. W. DeTemple, A quicker convergence to Euler's constant, Amer. Math. Monthly 100 (1993), no. 5, 468470. 
 In article  

[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, 103111. 
 In article  CrossRef 

[11]  B.N. Guo and F. Qi, Sharp bounds for harmonic numbers, Appl. Math. Comput. 218 (2011), no. 3, 991995. 
 In article  CrossRef 

[12]  B.N. Guo and F. Qi, Sharp inequalities for the psi function and harmonic numbers, Analysis (Berlin) 34 (2014), no. 2, 201208. 
 In article  CrossRef 

[13]  E. A. Karatsuba, On the computation of the Euler constant, Numer. Algor. 24 (2000), no. 12, 8397. 
 In article  CrossRef 

[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, 4550. 
 In article  CrossRef 

[15]  C. Mortici, A quicker convergence toward the constant with the logarithm term involving the constant e, Carpathian J. Math. 26 (2010), no. 1, 8691. 
 In article  

[16]  C. Mortici, Improved convergence towards generalized EulerMascheroni constant, Appl. Math. Comput. 215 (2010), no. 9, 34433448. 
 In article  CrossRef 

[17]  C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 59 (2010), no. 1, 97100. 
 In article  CrossRef 

[18]  C. Mortici, On new sequences converging towards the EulerMascheroni constant, Comput. Math. Appl. 59 (2010), no. 8, 26102614. 
 In article  CrossRef 

[19]  T. Negoi, A faster convergence to Euler's constant, Math. Gaz. 83 (1999), no. 498, 487489. 
 In article  

[20]  P. Paule and C. Schneider, Computer proofs of a new family of harmonic number identities, Adv. Appl. Math. 31 (2003), no. 2, 359378. 
 In article  CrossRef 

[21]  F. Qi, Complete monotonicity of functions involving the qtrigamma and qtetragamma functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. 109 (2015), in press. 
 In article  CrossRef 

[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, 303308. 
 In article  CrossRef 

[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, 159169. 
 In article  CrossRef 

[24]  A. Sîntǎmǎrian A generalization of Euler's constant, Numer. Algor. 46 (2007) no. 2, 141151. 
 In article  CrossRef 

[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. 
 In article  

[26]  R. M. Young, Euler's constant, Math. Gaz. 75 (1991), 187190. 
 In article  CrossRef 
