New approximation inequalities for circular functions.

In this paper, we obtain some improved exponential approximation inequalities for the functions ( sin x ) / x and sec ( x ) , and we prove them by using the properties of Bernoulli numbers and new criteria for the monotonicity of quotient of two power series.

Introduction

The following result is known as the Mitrinovic–Adamovic inequality [1, 2]: Nishizawa [3] gave the upper bound of the function in the form of the above inequality (1.1) and obtained the following power exponential inequality: Chen and Sándor [4] looked into the bounds for the function secx and obtain the following result for : Nishizawa [3] obtained the following inequality with power exponential functions derived from the right-hand inequality side of (1.3):

The purpose of this article is to establish some exponential approximation inequalities which improve the ones of (1.1)–(1.4). We prove these results for circular functions by using the properties of Bernoulli numbers and new criteria for the monotonicity of quotient of two power series.

Theorem 1.1

Let , and . Then we have where a and b are the best constants in (1.5).

Theorem 1.2

Let , and . Then we have where c and d are the best constants in (1.6).

Theorem 1.3

Let , and . Then we have where b and p are the best constants in (1.7).

Theorem 1.4

Let , Then we have where α and β are the best constants in (1.8).

We note that the right-hand side of the inequality (1.5) is stronger than that one in (1.2) due to while the double inequality (1.6) and (1.8) are sharper than the (1.5) and (1.7), respectively.

Lemmas

Lemma 2.1

([5-8])

Let be the even-indexed Bernoulli numbers,  . Then

Lemma 2.2

Let be the even-indexed Bernoulli numbers. Then the following power series expansion: and hold.

Proof

The following power series expansions can be found in [9, 1.3.1.4(2)(3)]: By (2.5) and (2.6) we have and  □

Lemma 2.3

([10])

Let and () be real numbers, and let the power series and be convergent for (). If for  , and if is strictly increasing (or decreasing) for  , then the function is strictly increasing (or decreasing) on ().

In order to prove Theorem 1.4, we need the following lemma. We introduce a useful auxiliary function . For , let f and g be differentiable on and on . Then the function is defined by The function has some good properties and plays an important role in the proof of a monotonicity criterion for the quotient of power series.

Lemma 2.4

([11])

Let and be two real power series converging on and for all k. Suppose that, for certain , the non-constant sequence is increasing (resp. decreasing) for and decreasing (resp. increasing) for . Then the function is strictly increasing (resp. decreasing) on if and only if . Moreover, if , then there exists such that the function is strictly increasing (resp. decreasing) on and strictly decreasing (resp. increasing) on .

Proof of Theorem 1.1

Let where and by Lemma 2.2. Let where

We now show that is increasing for . Since by Lemma 2.1, the proof of for can be completed when proving In fact, for . So is decreasing, and is decreasing on by Lemma 2.3. In view of , and , the proof of Theorem 1.1 is complete.

Proof of Theorem 1.2

(i) We first prove the left-hand side inequality of (1.6). Let Then by Lemma 2.2 we have where By Lemma 2.1, we have with due to , where It is not difficult to verify and for . So for , and for .

(ii) Then we prove the right-hand side inequality of (1.6). Let Then by Lemma 2.2 we have where By Lemma 2.1 we have that is, where for . So for and for .

(iii) Let Then This complete the proof of Theorem 1.2.

Proof of Theorem 1.3

(1) Let Then we get where We now show for , that is, or holds for . In fact, by Lemma 2.1 we have so (5.1) holds as long as we can prove that that is, which is equivalent to for . So for , which leads to , and is decreasing on . We can compute which give Then there exists an unique real number such that on and on . So is increasing on and decreasing on . Since there exists an unique real number such that on and on . So is increasing on and decreasing on . In view of , the proof of the left-hand side inequality of (1.7) is complete.

(2) Let Then we get where We now show for , that is, holds for . In fact, by Lemma 2.1 we have so (5.2) holds as long as we can prove that which is true for . So , and is increasing on . We can compute and , the proof of the right-hand side inequality of (1.7) is complete.

(3) Let Then this completes the proof of Theorem 1.3.

Proof of Theorem 1.4

Let where and with

Since and we can obtain but for . The inequality (6.1) is equivalent to By Lemma 2.1, we have and So (6.1) holds when we prove or which is ensured for .

So Since we see that is increasing on by Lemma 2.4. In view of the proof of Theorem 1.4 is complete.

Remark

Remark 7.1

The results of inequalities in Theorems 1.1–1.4 can be validated by methods and algorithms developed in [12, 13] and [14].