# Generalizations of Shafer-Fink-Type Inequalities for the Arc Sine Function

- Wenhai Pan
^{1}and - Ling Zhu
^{1}Email author

**2009**:705317

https://doi.org/10.1155/2009/705317

© W. Pan and L. Zhu. 2009

**Received: **29 December 2008

**Accepted: **28 April 2009

**Published: **19 May 2009

## Abstract

We give some generalizations of Shafer-Fink 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.

The theorem is generalized by Fink [2] as follows.

Theorem 1.2.

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.

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.

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 Shafer-Fink type double inequalities as follows.

Theorem 1.5.

holds, where is a point in and satisfies .

Theorem 1.6.

## 2. One Lemma: L'Hospital's Rule for Monotonicity

## 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

Since decreases on , we obtain that decreases on by using Lemma 2.1. At the same time, , , and , .

There are four cases to consider.

Since , decreases on , and , . So when and , (3.1) and (1.6) hold.

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.

Now, also firstly increases on then decreases on , and , . So when and , (3.1) and (1.6) hold too.

- (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.

Since , decreases on , and , . If and , then (3.1) holds on and (1.8) holds.

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.

Now, also firstly increases on then decreases on , and , . If and , then (3.1) holds on and (1.8) holds too.

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

## Authors’ Affiliations

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