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

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.

Let Then

(1.1)

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

Theorem 1.2.

Let Then

(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.

Let Then

(1.3)

Furthermore, 3 and and are the best constants in (1.3).

Malesevic [46] obtains the following inequality by using -method and computer separately.

Theorem 1.4.

Let Then

(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.

Let , and . If

(1.5)

then

(1.6)

holds, where is a point in and satisfies .

Theorem 1.6.

Let and If

(1.7)

then

(1.8)

holds, where is a point in and satisfies .

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

Lemma 2.1 (see [915]).

Let be two continuous functions which are differentiable and on If is increasing (or decreasing) on , then the functions and are also increasing (or decreasing) on

3. Proofs of Theorems 1.5 and 1.6

  1. (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

(3.1)

holds for

Let , we have

(3.2)

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.

  1. (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

(3.3)

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

(41)

Furthermore, and , and are the best constants in (4.1).

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Correspondence to Ling Zhu.

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Pan, W., Zhu, L. Generalizations of Shafer-Fink-Type Inequalities for the Arc Sine Function. J Inequal Appl 2009, 705317 (2009). https://doi.org/10.1155/2009/705317

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Keywords

  • Simple Proof
  • Sine Function
  • Differential Method
  • Double Inequality
  • Basic Differential Method