1. Introduction and Main Results

In [1], the following elementary problem was posed, showing that for ,

(1.1)

In [2], the following three proofs for the inequality (1.1) were provided.

Solution by Grinstein

Direct computation gives

(1.2)

where

(1.3)

Now is positive for all ; whence is an increasing function.

Since , it follows that for .

Solution by Marsh

It follows from that

(1.4)

The desired result is obtained directly upon integration of the latter inequality with respect to from to for .

Solution by Konhauser

The substitution transforms the given inequality into , which is a special case of an inequality discussed on [3, pages 105-106] .

It may be worthwhile to note that the inequality (1.1) is not collected in the authorized monographs [4, 5].

In [4, pages 288-289], the following inequalities for the arc tangent function are collected:

(1.5)
(1.6)
(1.7)

where . The inequality (1.5) is better than (1.7).

The aim of this paper is to sharpen and generalize inequalities (1.1) and (1.5).

Our results may be stated as the following theorems.

Theorem 1.1.

For , let

(1.8)

where is a real number.

(1)When , the function is strictly increasing on .

(2)When , the function is strictly decreasing on .

(3)When , the function has a unique minimum on .

As direct consequences of Theorem 1.1, the following inequalities may be derived.

Theorem 1.2.

For ,

(1.9)

For ,

(1.10)

For , the inequality (1.9) is reversed.

Moreover, the constants and in inequalities (1.9) and (1.10) are the best possible.

2. Remarks

Before proving our theorems, we give several remarks on them.

Remark 2.1.

The substitution may transform inequalities in (1.9) and (1.10) into some trigonometric inequalities.

Remark 2.2.

The inequality (1.1) is the special case of the left-hand side inequality in (1.9).

Remark 2.3.

The inequality (1.5) is the special case of the reversed version of the left hand-side inequality in (1.9).

Remark 2.4.

Let

(2.1)

for and . Direct computation gives

(2.2)

Hence,

(1)when the derivative is negative for ;

(2)when the derivative has a unique zero which is the unique maximum point of for .

Accordingly,

(1)when the function attains its maximum

(2.3)

(2)when the unique zero of equals

(2.4)

and the function attains its maximum

(2.5)

for .

In a word, the sharp lower bounds of (1.10) are

(2.6)
(2.7)

for .

Similarly, the sharp upper bound of (1.10) is

(2.8)

Remark 2.5.

Similar to the deduction of inequalities (2.6) and (2.7), the sharp versions of (1.9) and its reversion are

(2.9)
(2.10)

Remark 2.6.

It is easy to verify that the right-hand side inequalities in (2.9) and (2.10) are included in the inequality (2.8).

By the famous software Mathematica, it is revealed that the inequality (2.7) contains (2.6) and the left-hand side inequality in (2.9), and that the inequality (2.7) and the left-hand side inequality in (2.10) are not included in each other.

In conclusion, the following double inequality is the best accurate one:

(2.11)

where denotes.

Remark 2.7.

For possible applications of the double inequality (2.11) in the theory of approximations, the accuracy of bounds in (2.11) for the arc tangent function is described by Figures 1 and 2.

The upper curves in Figures 1 and 2 are, respectively, the graphs of the functions

(2.12)

and the lower curves in Figures 1 and 2 are, respectively, the graphs of the functions

(2.13)

on the interval , where denotes.

These two figures are plotted by the famous software Mathematica 7.0.

Figure 1
figure 1

The differences between terms in (2. 11).

Full size image
Figure 2
figure 2

The ratios between terms in (2. 11).

Full size image

Remark 2.8.

The approach below used in the proofs of Theorems 1.1 and 1.2 has been employed in [6–9].

Remark 2.9.

This paper is a revised version of the preprint [10].

3. Proofs of Theorems

Now we are in a position to prove our theorems.

Proof of Theorem 1.1.

Direct calculation gives

(3.1)

Let

(3.2)

then

(3.3)

and the function

(3.4)

has two zeros

(3.5)

Further differentiation yields

(3.6)

This means that the functions and are increasing on . From

(3.7)

it follows that

  1. (1)

    when or , the derivative is negative and the function is strictly decreasing on . From

    (3.8)

it is deduced that on . Accordingly,

(a)when , the derivative and the function is strictly increasing on ;

(b)when , the derivative is negative and the function is strictly decreasing on ;

  1. (2)

    when , the derivative is positive and the function is increasing on . By (3.8), it follows that the function is positive on . Thus, the derivative is positive and the function is strictly increasing on ;

  2. (3)

    when , the derivative has a unique zero which is a minimum of on . Hence, by the second limit in (3.8), it may be deduced that

(a)when , the function is negative on , so the derivative is also negative and the function is strictly decreasing on ;

(b)when , the function has a unique zero which is also a unique zero of the derivative , and so the function has a unique minimum of the function on .

On the other hand, the derivative can be rewritten as

(3.9)

and the function

(3.10)

satisfies

(3.11)

When , the derivative is positive and the function is strictly increasing on . Since , the function is positive, and so the derivative is positive, on for . Consequently, when , the function is strictly increasing on . The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2.

Direct calculation yields

(3.12)

By the increasing monotonicity in Theorem 1.1, it follows that for , which can be rewritten as (1.9) for . Similarly, the reversed version of the inequality (1.9) and the right-hand side inequality in (1.10) can be procured.

When , the unique minimum point of the function satisfies

(3.13)

and so the minimum of on is

(3.14)

where , as a result, the left-hand side inequality in (1.10) follows. The proof of Theorem 1.2 is complete.