Refined estimates and generalizations of inequalities related to the arctangent function and Shafer's inequality

In this paper we give some sharper refinements and generalizations of inequalities related to Shafer's inequality for the arctangent function, stated in Theorems 1, 2 and 4 in [1], by C. Mortici and H.M. Srivastava.


Introduction
Inverse trigonometric functions play an important role and have many applications in engineering [8], [9], [10] and [25]. In particular, the arctangent function and various related inequalities have been studied and effectively applied to problems in fundamental sciences and many areas of engineering, such as electronics, mechanics, and aeronautics [10], [20], [21]; see also [11].

Statement 3 (Theorem 4, [1])
For every x > 0, it is asserted that The main results of this paper are refined estimates and generalizations of the inequalities given in Statements 1, 2 and 3. Although the inequalities (2), (3) and (4) hold for x > 0, considering them in a neighborhood of zero is of primary importance, as noted in [1].

Main results
First, let us recall some well-known power series expansions that will be used in our proofs.
For |x| ≤ 1, For |x| ≤ 1, The following power series expansion holds: , with B(0) = 1, B(1) = − 1 3 , and for m ≥ 2: Power series coefficients are calculated by applying c's product to the power series expansions arising from the following transformation of the corresponding function: It is easy to prove that sequence {B (m)} m∈N0 for m ≥ 1 satisfies the recurrence equation:

Refinements of the inequalities in Statement 1
Before we proceed to Theorem 1, which represents an improvement and generalization of Statement 1, we need the following lemmas.
Proof. In the proof of this lemma we use the Wilf-Zeilberger method [3], [5] and [7]. (The same approach we used in [30].) The assertion is obviously true for m = 1.
Let m ≥ 2 and Then we have Further we have (13) β(m + 1) Consider now the sequence {S(m)} m∈N, m≥2 , where (14) where m ∈ N 0 and k ∈ N 0 . It is not hard to verify that functions F(m, k) and G(m, k) satisfy the following relation: If we sum both sides of (15) over all k ∈ N 0 , we get the following relation: Therefore from (13) and (16) we conclude that . * ) An algorithm for determining function G(m, k) for a given function F(m, k) is described in [3]. Note that the pair of discrete functions F(m, k), G(m, k) is the so-called Wilf-Zeilberger pair. Corollary 1 Given that the sequences {B(m)} m∈N0 and {β(m)} m∈N0 satisfy the same recurrence relation and as they agree for m = 0 and m = 1, we conclude that Let us introduce the notation: where C(0) = C(1) = 0, and for m ≥ 2 the following holds: Thus, we have the power series expansion: (20) f Let us introduce the notation:

Lemma 2
For m ∈ N 0 the following holds: Proof. .

Lemma 3
For m ∈ N 0 the following holds: Proof. The statement immediately follows from the inequalities:

Lemma 4
For m ∈ N 0 the following holds: Proof.
Theorem 1 For the real analytic function: the following inequalities hold for k ∈ N and x ∈ 0, where C(0) = C(1) = 0, and for m ≥ 2 the following holds: Proof. We will prove that the sequence {C(m)} m∈N0 is positive, monotonically decreasing and tends to zero as m tends to infinity. We will use Lemmas 3 and 4.
Let us now prove that {C(m)} m∈N0 is a monotonically decreasing sequence.
It is easy to prove that C(m + 1) − C(m) < 0 for m ≥ 8, i.e. the sequence is monotonically decreasing. Since {C (m)} m∈N0 is positive for m ≥ 2, monotonically decreasing (for m ≥ 8), and tends to zero, the same holds true for the sequence C(m)x 2m+1 m∈N0 for a fixed x ∈ 0, √ 3/2 noting that it is decreasing for m ≥ 3 , so we can apply Leibniz's theorem for alternating series [6], thus proving the claim of Theorem 1:

Refinements of the inequalities in Statement 2
We propose the following improvement and generalization of Statement 2: Theorem 2 For every x ∈ (0, 1] and k ∈ N , it is asserted that Examples For x ∈ (0, 1] and k = 1 we get the inequality (3) from Statement 2. For x ∈ (0, 1] and k ≥ 2 the inequality (23) refines the inequality (3) from Statement 2 and we have the following new results: • Taking k = 2 in (23) • Taking k = 3 in (23) etc.
We prove that sequence {e(m)} m∈N, m≥2 is a monotonically decreasing sequence and lim m→+∞ e(m) = 0.
By the principle of mathematical induction, it follows that is true for all j ∈ N. Therefore S(m) > 0 for m ≥ 2, i.e. (29) e(m) > 0, for m ≥ 2.
Consider the following equivalences for m ≥ 2 : Consider the last inequality. It is easy to verify that it is true for m = 2. Finally, based on (28) we conclude that {E (m)} m∈N,m≥2 is a positive monotonically decreasing sequence, and that it tends to zero. The same holds true for the sequence E(m)x 2m+1 m∈N,m≥2 for a fixed x ∈ (0, 1] so we can apply Leibniz's theorem for alternating series [6], thus proving the claim of Theorem 2.

Refinements of the inequalities in Statement 3
We propose the following improvement and generalization of Statement 3: Theorem 3 For every x ∈ (0, 1] and k ∈ N , it is asserted that Examples For x ∈ (0, 1] and k = 1 we get the inequality (4) from Statement 3. For x ∈ (0, 1] and k ≥ 2 the inequality (33) from Theorem 3 refines the inequality (4) from Statement 3 and we have the following new results: • Taking k = 2 in (33) gives − 1 12 x 5 − 29 448 x 7 + 65 1152 x 9 .

Proof of Theorem 3.
For x ∈ (0, 1] the following power series expansion holds: We prove that the sequence {C(m)} m∈N is positive, monotonically decreasing and tends to zero as m tends to infinity.
As it is easy to show (by the principle of mathematical induction) that the last inequality holds true for m ∈ N , we may conclude that {C(m)} m∈N is a monotonically decreasing sequence.
Since {C (m)} m∈N is a positive monotonically decreasing sequence, and it tends to zero, the same holds true for the sequence C(m)x 2m+1 m∈N for a fixed x ∈ (0, 1]. So we can apply Leibniz's theorem for alternating series [6] and thus prove the claim of Theorem 3.

Conclusion
In Theorems 1, 2 and 3 of this paper we proved some new inequalities related to Shafer's inequality for the arctangent function. These inequalities represent sharpening and generalization of the inequalities given in [1] (Theorems 1, 2 and 4).
Let us mention that it is possible to prove the inequality (22), for any fixed k ∈ N and x ∈ 0, √ 3 2 , by substituting x = tan t for t ∈ 0, arctan √ 3 2 using the algorithms and methods * * ) developed in [28] and [29]. Also, the inequalities (23) and (33) for any fixed k ∈ N and x ∈ (0, 1] can be proved by substituting x = tan t for t ∈ 0, π