Monotonicity results and bounds for the inverse hyperbolic sine

In this note, we present monotonicity results of a function involving to the inverse hyperbolic sine. From these, we derive some inequalities for bounding the inverse hyperbolic sine.


Introduction and main results
In [2, Theorem 1.9 and Theorem 1.10], the following inequalities were established: For 0 ≤ x ≤ r and r > 0, the double inequality holds true if and only if a ≤ 2 and b ≥ √ 1 + r 2 arcsinh r − r r − arcsinh r .
The aim of this paper is to elementarily generalize the inequality (1) to monotonicity results and to deduce more inequalities.
Our results may be stated as the following theorems.
As straightforward consequences of Theorem 1, the following inequalities are inferred.
(2) For θ > 2, the double inequality Remark 1. Replacing arcsinh x by x in (4) and (5) yields for x ∈ (0, arcsinh r). This can be regarded as Oppenheim type inequalities for the hyperbolic sine and cosine functions. For information on Oppenheim's double inequality for the sine and cosine functions, please refer to [1] and closely related references therein.
Remark 2. It is clear that the left-hand side inequality in (4) recovers the left-hand side inequality in (1) while the right-hand side inequalities in (1) and (4) do not include each other.
Remark 3. By similar approach to prove our theorems in next section, we can procure similar monotonicity results and inequalities for the inverse hyperbolic cosine.

Proof of theorems
Now we prove our theorems elementarily.
For θ > 2, the minimum point x 0 ∈ (0, ∞) satisfies Therefore, the minimum of the function f θ (x) on (0, ∞) equals From this, it is obtained that for x ∈ (0, r], which implies the inequality (5). The proof of Theorem 2 is thus completed.