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

(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 well-known 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.

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