Generalizations of Shafer-Fink-Type Inequalities for the Arc Sine Function
© W. Pan and L. Zhu. 2009
Received: 29 December 2008
Accepted: 28 April 2009
Published: 19 May 2009
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.
Shafer (see Mitrinovic and Vasic [1, page 247]) gives us a result as follows.
The theorem is generalized by Fink  as follows.
In , Zhu presents an upper bound for and proves the following result.
In this paper, we give some generalizations of these above results and obtain two new Shafer-Fink type double inequalities as follows.
2. One Lemma: L'Hospital's Rule for Monotonicity
3. Proofs of Theorems 1.5 and 1.6
We first process the proof of Theorem 1.5.
There are four cases to consider.
Now we consider proving Theorem 1.6.
There are four cases to consider too.
4. The Special Cases of Theorems 1.5 and 1.6
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