Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 4178629, 8 pages
https://doi.org/10.1155/2018/4178629

Research Article

Refined Estimates and Generalizations of Inequalities Related to the Arctangent Function and Shafer’s Inequality

Branko Malešević, Marija Rašajski, and Tatjana Lutovac

University of Belgrade, School of Electrical Engineering, Department of Applied Mathematics, Serbia

Correspondence should be addressed to Branko Malešević; branko.malesevic@etf.bg.ac.rs

Received 10 November 2017; Accepted 11 September 2018; Published 10 October 2018

Academic Editor: Luis M. López-Ochoa

Copyright © 2018 Branko Malešević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We give some sharper refinements and generalizations of inequalities related to Shafer’s inequality for the arctangent function, stated in   , ,  and   in Mortici and Srivastava, 2014, by C. Mortici and H.M. Srivastava.

1. Introduction

Inverse trigonometric functions play an important role and have many applications in engineering [1–4]. 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 [3, 5, 6]; see also [7].

Various approximations of the arctangent function can be found in [4–6, 8–21]; see also [22, 23]. One of the inequalities that attracted attention of many authors is Shafer’s inequality [9]:

which holds for ; see also [10–12].

Recently, in [8], Mortici and Srivastava proved the following results, cited here as Statements 1, 2, and 3, related to the above inequality. These results are the starting point of our research.

Statement 1 (Theorem 1, [8]). For every , the following two-sided inequality holds

where and .

Statement 2 (Theorem 2, [8]). For every , it is asserted that

where and .

Statement 3 (Theorem 4, [8]). For every , 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 inequalities (2), (3), and (4) hold for , considering them in a neighborhood of zero is of primary importance, as noted in [8].

2. Main Results

First, let us recall some well-known power series expansions that will be used in our proofs.

For ,

where

For ,

where

The following power series expansion holds:

where , with and , and for ,

Power series coefficients are calculated by applying Cauchy’s product to the power series expansions arising from the following transformation of the corresponding function:

It is easy to prove that sequence for satisfies the recurrence equation:

2.1. Refinements of the Inequalities in Statement 1

Before we proceed to Theorem 9, which represents an improvement and generalization of Statement 1, we need the following lemmas.

Lemma 4. Let , , and , for . The sequence for satisfies the recurrence relation (12).

Proof. In the proof of this lemma we use the Wilf- Zeilberger method [24–26]. (The same approach we used in [27].)
The assertion is obviously true for .
Let and

Then we have

Further we have

Consider now the sequence , where

Consider the function (an algorithm for determining function for a given function is described in [24]. Note that the pair of discrete functions is the so-called Wilf- Zeilberger pair)

where and . It is not hard to verify that functions and satisfy the following relation:

If we sum both sides of (18) over all , we get the following relation:

Finally, as

we have

Therefore from (15) and (21) we conclude that

Corollary 5. Given that the sequences and satisfy the same recurrence relation and as they agree for and , we conclude that

Let us introduce the notation:

where , and for the following holds:

Thus, we have the power series expansion:

for and .

Let us introduce the notation:

Lemma 6. For the following holds:

Proof.

Lemma 7. For the following holds:

Proof. The statement immediately follows from the inequalities:

Lemma 8. For the following holds:

Proof.

Theorem 9. For the real analytic function

the following inequalities hold for and :

where , and for the following holds:

Proof. We will prove that the sequence is positive and monotonically decreasing and tends to zero as tends to infinity. We will use Lemmas 7 and 8.

It is easy to verify that for ; therefore . Let us note that , so we can conclude that .
Let us now prove that is a monotonically decreasing sequence.

It is easy to prove that for , i.e., the sequence is monotonically decreasing. Since is positive for and monotonically decreasing (for ) and tends to zero, the same holds true for the sequence for a fixed (noting that it is decreasing for ), so we can apply Leibniz’s theorem for alternating series [28], thus proving the claim of Theorem 9:

Examples. For and we get Statement 1.

For and ,

For and ,

etc.

2.2. Refinements of the Inequalities in Statement 2

We propose the following improvement and generalization of Statement 2.

Theorem 10. For every and , it is asserted that

where

Examples. For and we get inequality (3) from Statement 2.

For and inequality (42) refines inequality (3) from Statement 2 and we have the following new results:

(i)Taking in (42) gives

(ii)Taking in (42) gives

 etc.

Proof of Theorem 10. Based on Cauchy’s product of power series (7) and (5), the real analytical function,

for has the following power series:

where

We aim to show that sequence decreases monotonically and that . It is easy to verify that sequence satisfies the following recurrence relation:

Consider the sequence where

and

and

It is easy to verify that sequence satisfies the recurrence relation (49). Given that sequences and agree for and , we conclude that

We prove that sequence is a monotonically decreasing sequence and .
By the principle of mathematical induction, it follows that

is true for all Therefore for , i.e.,

To prove that is a monotonically decreasing sequence, let us use the following notation:

where

Consider the following equivalences for :

Consider the last inequality. It is easy to verify that it is true for . Observing that

and using the induction hypothesis for some positive integer , we conclude that

Therefore, by the principle of mathematical induction, the inequality

is true for , i.e.,

Let us further consider the positive addend of , i.e.,

By the principle of mathematical induction, it follows that

Finally, given that for

we have
Finally, based on (53) we conclude that is a positive monotonically decreasing sequence and that it tends to zero. The same holds true for the sequence for a fixed so we can apply Leibniz’s theorem for alternating series [28], thus proving the claim of Theorem 10.

2.3. Refinements of the Inequalities in Statement 3

We propose the following improvement and generalization of Statement 3.

Theorem 11. For every and , it is asserted that

where

Examples. For and we get inequality (4) from Statement 3.

For and inequality (66) from Theorem 11 refines inequality (4) from Statement 3 and we have the following new results:

(i)Taking in (66) gives

(ii)Taking in (66) gives

 etc.

Proof of Theorem 11. For the following power series expansion holds:

where

We prove that the sequence is positive and monotonically decreasing and tends to zero as tends to infinity.
It is easy to verify that is true for . Thus, the following equivalences hold true for every :

and we conclude that for every .
Let us now prove that is a monotonically decreasing sequence. We have

As it is easy to show (by the principle of mathematical induction) that the last inequality holds true for , we may conclude that is a monotonically decreasing sequence.
Finally, as , we conclude that .
Since is a positive monotonically decreasing sequence, and it tends to zero, the same holds true for the sequence for a fixed . So we can apply Leibniz’s theorem for alternating series [28] and thus prove the claim of Theorem 11.

3. Conclusion

In Theorems 9, 10, and 11 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 [8] ().

Let us mention that it is possible to prove inequality (35), for any fixed and , by substituting for using the algorithms and methods (see also [29, 30]) developed in [31, 32]. Also, inequalities (42) and (66) for any fixed and can be proved by substituting for using the algorithms and methods (see also [29, 30]) developed in [31, 32].

Conflicts of Interest

The authors would like to state that they do not have any conflicts of interest in the subject of this research.

Authors’ Contributions

All the authors participated in every phase of the research conducted for this paper.

Acknowledgments

Research of the first, second, and third author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under Projects ON 174032, III 44006, ON 174033, and TR 32023, respectively.

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