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Generalizations of ShaferFinkType Inequalities for the Arc Sine Function
Journal of Inequalities and Applications volume 2009, Article number: 705317 (2009)
Abstract
We give some generalizations of ShaferFink inequalities, and prove these inequalities by using a basic differential method and l'Hospital's rule for monotonicity.
1. Introduction
Shafer (see Mitrinovic and Vasic [1, page 247]) gives us a result as follows.
Theorem 1.1.
Let Then
The theorem is generalized by Fink [2] as follows.
Theorem 1.2.
Let Then
Furthermore, 3 and are the best constants in (1.2).
In [3], Zhu presents an upper bound for and proves the following result.
Theorem 1.3.
Let Then
Furthermore, 3 and and are the best constants in (1.3).
Malesevic [4–6] obtains the following inequality by using method and computer separately.
Theorem 1.4.
Let Then
Zhu [7, 8] offers some new simple proofs of inequality (1.4) by L'Hospital's rule for monotonicity.
In this paper, we give some generalizations of these above results and obtain two new ShaferFink type double inequalities as follows.
Theorem 1.5.
Let , and . If
then
holds, where is a point in and satisfies .
Theorem 1.6.
Let and If
then
holds, where is a point in and satisfies .
3. Proofs of Theorems 1.5 and 1.6

(A)
We first process the proof of Theorem 1.5.
Let for , in which case the proof of Theorem 1.5 can be completed when proving that the double inequality
holds for
Let , we have
where and , , , .
Since decreases on , we obtain that decreases on by using Lemma 2.1. At the same time, , , and , .
There are four cases to consider.
Case ()
Since , decreases on , and , . So when and , (3.1) and (1.6) hold.
Case ()
At this moment, there exists a number such that , is positive on and negative on . That is, firstly increases on then decreases on , and , . So when and , (3.1) and (1.6) hold.
Case ()
Now, also firstly increases on then decreases on , and , . So when and , (3.1) and (1.6) hold too.
Case (
Since , increases on , , and . So when and , (3.1) and (1.6) hold.

(B)
Now we consider proving Theorem 1.6.
In view of the fact that (1.8) holds for , we suppose that in the following.
First, let and for , we have and . Second, let , then and (1.8) is equivalent to
When letting and (), (3.3) becomes (3.1).
Let . At this moment, decreases on , , , and , .
There are four cases to consider too.
Case ()
Since , decreases on , and , . If and , then (3.1) holds on and (1.8) holds.
Case (
At this moment, there exists a number such that , is positive on and negative on . That is, firstly increases on then decreases on , and , . If and , then (3.1) holds on and (1.8) holds.
Case (
Now, also firstly increases on then decreases on , and , . If and , then (3.1) holds on and (1.8) holds too.
Case (
Since , increases on , , and . If and , then (3.1) holds on and (1.8) holds.
4. The Special Cases of Theorems 1.5 and 1.6
(1)Taking in Theorem 1.5 and in Theorem 1.6 leads to the inequality (1.1).
(2)Taking in Theorem 1.5 and in Theorem 1.6 leads to the inequality (1.4).
(3)Let in Theorem 1.5 and in Theorem 1.6, we have the following result.
Theorem 4.1.
Let . Then
Furthermore, and , and are the best constants in (4.1).
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Pan, W., Zhu, L. Generalizations of ShaferFinkType Inequalities for the Arc Sine Function. J Inequal Appl 2009, 705317 (2009). https://doi.org/10.1155/2009/705317
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DOI: https://doi.org/10.1155/2009/705317
Keywords
 Simple Proof
 Sine Function
 Differential Method
 Double Inequality
 Basic Differential Method