Padé approximants for inverse trigonometric functions and their applications.

The Padé approximation is a useful method for creating new inequalities and improving certain inequalities. In this paper we use the Padé approximant to give the refinements of some remarkable inequalities involving inverse trigonometric functions, it is shown that the new inequalities presented in this paper are more refined than that obtained in earlier papers.

Introduction

The inequality referred to as the Wilker inequality, is one of the most famous inequalities for trigonometric functions.

The inverse trigonometric version of the Wilker inequality was considered in [1] and [2], as follows:

The Shafer-Fink double inequality for the arctangent function asserts that holds for any positive real number x.

Shafer’s inequality was recently generalized by Qi et al. in [3], as follows: where and .

The Shafer-Fink double inequality for the arcsine function states that holds for . Furthermore, 3 and π are the best constants in (1.5).

For more relevant papers on the above topic, we refer the reader to [4-16] and the references therein.

It should be noted that these inequalities were proved using the variation of some functions and their derivatives, meanwhile, some of the above inequalities were improved using Taylor’s expansions of inverse trigonometric functions.

The Padé approximant is of the form of one polynomial divided by another polynomial, the technique was developed around 1890 by Henri Padé. It is well known that a Padé approximant is the ‘best’ approximation of a function by a rational function of given order. The rational approximation is particularly good for series with alternating terms and poor polynomial convergence (see [17-19]).

It is the aim of this paper to give the refinements of the aforesaid inverse trigonometric inequalities using the Padé approximant method.

Suppose that we are given a power series for representing a function , i.e.,

For a given function , the Padé approximant of order is the rational function which agrees with at 0 to the highest possible order, i.e.,

One can prove that the Padé approximant of a function is unique for given m and n, that is, the coefficients , can be uniquely determined.

In fact, with the help of the Taylor series, the Padé approximant of can be derived from the following relationship: where is a polynomial factor.

For example, we consider the Taylor series and its associate polynomial .

Then the Padé approximant of arcsinx for the order is written as which satisfies

From the above equation, we find

Therefore

In Table 1 we provide a list of Padé approximants for arcsinx and arctanx which will be used in subsequent sections.

Table 1

Padé approximants for arcsin and arctan

Function	Padé approximant	Associate Taylor polynomials
arcsinx	arcsin_[1/2](x)=66−x^2	x+x^36
arcsinx	arcsin_[5/2](x)=61x^5+1,080x^3−2,520x1,500x^2−2,520	x+x^36+3x^540+5x^7112
arctanx	arctan_[1/2](x)=3xx^2+3	x−x^33
arctanx	arctan_[3/2](x)=4x^3+15x9x^2+15	x−x^33+x^55
arctanx	arctan_[5/2](x)=−4x^5+40x^3+105x75x^2+105	x−x^33+x^55−x^77
arctanx	arctan_[3/4](x)=55x^3+105x9x^4+90x^2+105	x−x^33+x^55−x^77

Some lemmas

In order to prove the main results in Section 3, we need the following lemmas.

Lemma 2.1

For every , one has

Proof

We consider the function

Differentiating with respect to x yields

Evidently, for . Then is strictly increasing on . As , we get for , this proves the validity of inequality (2.1). The proof of Lemma 2.1 is complete. □

Lemma 2.2

For every , one has

We introduce a function ,

Its derivative is positive for every , therefore is strictly increasing on . As , we have for , which implies the desired inequality (2.2). Lemma 2.2 is proved. □

Lemma 2.3

For any positive real number x, one has

We define a function by

Then we have for , thus ψ is strictly decreasing on . It follows from that for .

Let

Differentiating with respect to x gives

We have for , hence ϕ is strictly increasing on . Since , we deduce that for .

The inequality (2.3) is proved. This completes the proof of Lemma 2.3. □

Main results

In this section we will formulate and prove the refinements of the aforesaid inverse trigonometric inequalities.

Theorem 3.1

For every , the following inequality holds: where .

We remark that if the above inequality is true for , then it holds clearly for . Therefore, it is sufficient to prove only for .

By using Lemmas 2.1 and 2.2, we have

Additionally, which implies the desired inequality (3.1). The proof of Theorem (3.1) is completed. □

Remark 1

By making use of the Taylor expansion, we have

We note that for . Hence the Padé approximation provides a better inequality than the inequality below

The Taylor’s approximation prompts to consider another method for proving the inequality asserted in Theorem 3.1, that is, we may add more terms to the Taylor polynomial (3.2) in order to prove the inequality (3.1). However, in this way, we will face complicated calculations on the high-order derivatives of and some cumbersome inequalities resulting from it.

Theorem 3.2

Let . Then for we have the following inequality:

The first inequality can be rewritten as

Since , it is easy to find that

Thus we need to prove the following inequality: or, equivalently, which can be deduced from a simple calculation:

Therefore, we obtain

The last inequality can be rewritten as

Since , it is easy to observe that

Hence we need to prove the following inequality: or, equivalently,

Now, from together with we conclude that which leads to the inequality

In conclusion, the inequality (3.4) follows immediately from inequalities (3.5), (3.6), and the inequality (2.3) given by Lemma 2.3. This completes the proof of Theorem 3.2. □

Finally, we deal with the improved version of the Shafer-Fink inequality (1.5).

Using the well-known trigonometric identity and the double inequality from the Lemma 2.3, we obtain

Substituting the expression by y (), we get

After some elementary calculations, the above inequality can be transformed to the following refinement of Shafer’s inequality for the arcsine function.

Theorem 3.3

For every , one has

For the sake of completeness, besides the solution above, we will provide another elementary solution in regard to the inequalities (3.7).

Putting , in (3.7) gives

The left-hand side inequality of (3.8) can be rewritten as

Let us consider the function ,

Its derivatives are and

Using the formula the function can be further rearranged as

Then

Note that

Therefore is strictly increasing on . As , it follows that for . Hence for .

Using similar arguments, we have for . The left-hand side inequality of (3.8) is proved.

In order to prove the right-hand side inequality of (3.8), we observe that

It is easy to find that the right-hand side inequality of (3.8) is equivalent to the following inequality:

Define a function by

Differentiating with respect to t gives

The function has the derivative

Also, we have

Since , it follows that for . Therefore for .

Using the same arguments, we have for , the right-hand side inequality of (3.8) is proved. The proof of Theorem 3.3 is completed. □

Remark 2

It is easy to observe that

Thus, the left-hand side inequality of (3.7) is stronger than the left-hand side inequality of (1.5).

Also, we have and where

Hence we conclude that holds for , which implies that the right-hand side inequality of (3.7) is stronger than the right-hand side inequality of (1.5) when .

Conclusions

The Padé approximation is a useful method for creating new inequalities and improving certain inequalities. Firstly, we introduce the main technique of Padé approximation and establish some Padé approximants for arcsinx and arctanx. And then we use the Padé approximation to improve some remarkable inequalities involving inverse trigonometric functions, we show that the new inequalities presented in this paper are more refined than that obtained in earlier papers. It is worth to mention that the Padé approximation method has also been applied to dealing with the refinements of certain inequalities for trigonometric functions and hyperbolic functions in our recent papers [20] and [21]. We expect that the method will be useful in solving some others problems concerning inequalities.