Research Article | Open Access
Refined Estimates and Generalizations of Inequalities Related to the Arctangent Function and Shafer’s Inequality
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.
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 .
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 :which holds for ; see also [10–12].
Statement 1 (Theorem 1, ). For every , the following two-sided inequality holdswhere and .
Statement 2 (Theorem 2, ). For every , it is asserted thatwhere and .
Statement 3 (Theorem 4, ). 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 .
2. Main Results
First, let us recall some well-known power series expansions that will be used in our proofs.
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
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 .)
The assertion is obviously true for .
Let and Then we have Further we haveConsider now the sequence , whereConsider the function (an algorithm for determining function for a given function is described in . 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 haveTherefore 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:
Lemma 7. For the following holds:
Proof. The statement immediately follows from the inequalities:
Lemma 8. For the following holds:
Theorem 9. For the real analytic functionthe 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 , 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 thatwhere
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:whereWe 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 thatWe 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 thatand 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 thatFinally, 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 , 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 thatwhere
Proof of Theorem 11. For the following power series expansion holds: whereWe 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  and thus prove the claim of Theorem 11.
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  ().
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.
All the authors participated in every phase of the research conducted for this paper.
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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