Some New Inequalities of Jordan Type for Sine
The authors find some new inequalities of Jordan type for the sine function. These newly established inequalities are of new form and are applied to deduce some known results.
For , we have The inequality is sharp with equality if and only if . This inequality is known in the literature as Jordan’s inequality for the sine function. See [1, page 33] and other references cited in the first page of .
Motivated by , it was established in  that for . The equalities are valid if and only if . This also refines Jordan’s inequality (1). Also, see the double inequality (3.10) in the survey article [2, page 17].
In recent years, the above inequalities have been refined, extended, generalized, and applied by many mathematicians in a large amount of papers. See, for example, [3–19]. For a systematic review on this topic, please refer to the expository paper .
The aim of this paper is to further refine and generalize these inequalities of Jordan type for the sine function.
Our main results may be stated as in the following theorems.
Theorem 1. If and are integers, then on .
Theorem 2. Suppose that is a -time differentiable function on . If the function satisfies and on , then
Remark 3. Taking in Theorem 1 yields
on for . The equalities in (7) are valid if and only if .
Putting in (7) results in (2) and (3), respectively. This means that Theorem 1 generalizes the inequalities (2) and (3).
In the final section of this paper, we will apply Theorem 1 to refine and generalize Yang’s inequality and construct some integral inequalities.
2. A Lemma
Lemma 5. Let be differentiable on . If and are decreasing on , then the functions are also decreasing on .
3. Proofs of Theorems
We are now in a position to prove our theorems.
Proof of Theorem 1. Let on . A direct calculation gives where Utilizing on leads to on for and . As a result, the function is decreasing on . In virtue of Lemma 5, it follows that the functions are all decreasing on . Since we have which can be reformulated as the inequality (4). Theorem 1 is thus proved.
Proof of Theorem 2. Let on . It is easy to see that on . Furthermore, we have Employing and the conditions in (5), it is not difficult to show that the numerator of is negative on . This means that the function is decreasing on . Consequently, making use of Lemma 5 consecutively, it is revealed that the functions are all decreasing on . Since from , the inequality (6) follows. The proof of Theorem 2 is complete.
4. Applications of Theorem 1
Let and with . Then, This inequality is known in the literature as Yang’s inequality. Since paper , many mathematicians mistakenly referred this inequality to [21, pages 116–118]. Indeed, the paper we should refer to is  or an even earlier paper in Chinese.
Theorem 7. For , let and . If , then where and and are integers.
Proof. Substituting in the inequality (4) reveals that Using the inequality see either , [16, ], or [2, page 17, ], becomes Finally, taking the sum of the above inequality for all results in (24). The required proof is complete.
Corollary 8. Under the conditions of Theorem 7, one has where
Proof. When and , we have This implies the required result.
The second application of Theorem 1 is to construct some new integral inequalities for .
Theorem 9. For , if and are integers, then
Proof. This follows from integrating on all sides of the double inequality (4).
Remark 10. Applying Theorem 9 to gives Applying Theorem 9 to and yields This is a recovery of an inequality established in [3, page 101]. It was also collected in [2, ]. Such a kind of inequalities can be found in .
The authors appreciate Professor Dr. Feng Qi (F. Qi) at Tianjin Polytechnic University in China for his kind and valuable contributions to this paper.
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