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 a systematic review on this topic, please refer to the expository paper [2].
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 ≥ 0 and ≥ 2 are integers, then 2 The Scientific World Journal on (0, /2], then Remark 3. Taking = 0 in Theorem 1 yields on (0, /2] for ≥ 2. The equalities in (7) are valid if and only if = /2.
In the final section of this paper, we will apply Theorem 1 to refine and generalize Yang's inequality and construct some integral inequalities.

A Lemma
In order to prove Theorems 1 and 2, the following lemma is necessary.

Proofs of Theorems
We are now in a position to prove our theorems.

Applications of Theorem 1
After proving Theorems 1 and 2, we now start off to apply them to construct some new inequalities. Let 0 ≤ ≤ 1 and , > 0 with + ≤ . Then, 4 The Scientific World Journal This inequality is known in the literature as Yang's inequality. Since paper [16], many mathematicians mistakenly referred this inequality to [21, pages 116-118]. Indeed, the paper we should refer to is [22] or an even earlier paper in Chinese. The first application of Theorem 1 is to refine and generalize Yang's inequality (23) as follows.
Proof. Substituting = /2 in the inequality (4) reveals that Using the inequality see either [22], [ Finally, taking the sum of the above inequality for all 1 ≤ < ≤ results in (24). The required proof is complete.

Corollary 8. Under the conditions of Theorem 7, one has
Proof. When 0 ≤ < 1 and = 0, we have This implies the required result.
The second application of Theorem 1 is to construct some new integral inequalities for sin / .
Proof. This follows from integrating on all sides of the double inequality (4). It was also collected in [2, (2.14)]. Such a kind of inequalities can be found in [23].