Sharp refined quadratic estimations of Shafer's inequalities

1.   Introduction

Shafer ^[1,2,3] established the following result for arctangent function, which is known as Shafer's inequality:

Theorem 1. Let $x > 0$. Then

holds.

The author of this paper ^[4] derived an upper bound for $\arctan x$, and obtained the following result.

Theorem 2. (The double Shafer inequality, [4, Theorem 3]) Let $x > 0$. Then

Furthermore, $80/3$ and $256/\pi ^{2}$ are the best constants in $(1.2)$.

During the past few years, the inequalities on inverse trigonometric functions have been the subject of intensive research. In particular, some interesting generalizations and improvements relating to (1.1) and (1.2) can be found in the literature ^{[5,6,7,8,9,10,11,12,13]}. Using the main result of ^[14], Nishizawa ^[5] gave the following conclusion, which does not contain (1.2).

Theorem 3. ([5, Theorem 2.1]) For $x > 0$, we have

where the constants $(\pi ^{2}-4)^{2}$ and $32$ are the best possible.

In this paper, using the power series expansions of $(\tan x)^{k}(k = 1, 2, 3)$ and the monotonicity of a function involving the Riemann's zeta function, we sharpen the quadratic estimations of Shafer's inequality (1.3) as follows.

Theorem 4. Let $p$ $> 0$. Then we have

(ⅰ) the inequality

holds for all $x\in (0, \infty)$ if and only if $p\in \lbrack (\pi ^{2}-4)^{2}, \infty)$;

(ⅱ) the inequality

holds for all $x\in (0, \infty)$ if $p\in (0, 32]$.

Theorem 5. The inequality

holds for all $x\in (0, \infty)$ if and only if $p\in (32, \infty)$.

Theorem 6. The inequality

holds for all $x\in (0, \infty)$ if $p\in (0, 16+4\pi ^{2}/3]$.

Obviously, Theorem 3 is a straightforward consequence of Theorem 4.

2.   Lemmas

Lemma 1. ([14, Lemma 2.1; 15, Lemma 2.1]) The function

is decreasing, where $\zeta (n)$ is Riemann's zeta function.

Lemma 2. ([16, Theorem 3.4]) Let $\zeta (n)$ be Riemann's zeta function and $B_{2n}$ the even-indexed Bernoulli numbers. Then

Lemma 3. Let $B_{2n}$ be the even-indexed Bernoulli numbers. Then for $n = 1, 2, \cdots,$

holds.

Proof. Since $(1-1/2^{2n})\zeta (2n)$ is decreasing by Lemma $1$, it follows that

Using the representation $(2.2),$ the inequality $(2.4)$ becomes

that is, the inequality $(2.3)$ holds.

Lemma 4. Let $\left\vert t\right\vert < \pi /2$. Then

hold.

Proof. From [17, p.133] we have

Then we obtan

and

where

with $u(1) = 0$. This completes the proof of Lemma $4$.

Lemma 5. The function

is decreasing on $(0, \pi /2)$.

Proof. From $(2.6)$–$(2.8)$, we have

where

By Lemma $3$ and $(2.10)$ we get that $q(n) > 0$ for $n = 2, 3, \cdots.$ This leads to

so $g(t)$ is decreasing on $(0, \pi /2)$.

Lemma 6. Let $t\in (0, \pi /2),$ $p > 32$. Then

holds if and only if

Proof. The inequality $(2.12)$ is equivalent to

Let

and

Then

and the inequality $(2.14)$ is equivalent to

where

Since for $0 < t < \pi /2$,

from $(2.15)$ we have

that is, the inequality $(2.13)$ is true.

3.   Proofs of Theorems 4–6

Let $t = \arctan x$. Then $t\in (0, \pi /2)$. We investigate the maximum and minimum values of the function

on the interval $(0, \pi /2)$.

We can compute

where

By Lemma $5$ we obtain that

We consider the following three cases.

Case 1: When $p\geq \max_{t\in (0, \pi /2)}g(t) = \left(\pi ^{2}-4\right) ^{2} \simeq 34.\, 452,$ we have $G^{\prime }(t)\leq 0$, and the function $G(t)$ is decreasing on $(0, \pi /2)$. In view of

we obtain

Since the three functions $\pi ^{2}\tan t-4t$, $t$, and $p+(2\pi \tan t)^{2}$ are positive on $(0, \pi /2)$, the right-hand side inequality of $(3.1)$ leads to $(1.4)$ while the left-hand side one leads to $(1.6)$ by Lemma $6$.

Case 2: When $p\leq \min_{t\in (0, \pi /2)}g(t) = 4\pi ^{2}/3+16 \simeq 29.\, 159,$ we have $G^{\prime }(t)\geq 0$, so the function $G(t)$ is increasing on $(0, \pi /2)$. Then

The left-hand side inequality of $(3.2)$ leads to $(1.5)$ while the right-hand side one is the inequality $(1.7)$.

Case 3: When $4\pi ^{2}/3+16 < p < \left(\pi ^{2}-4\right) ^{2},$ we set $g(t)-p: = q(t)$. Since $q(0^{+}) = g(0^{+})-p = \left(\pi ^{2}-4\right) ^{2}-p > 0$ and $q((\pi /2)^{-}) = g((\pi /2)^{-})-p = 4\pi ^{2}/3+16 -p < 0$, there is a unique point $\xi \in (0, \pi /2)$ such that $G^{\prime }(t) > 0$ for all $t\in (0, \xi)$ and $G^{\prime }(t) < 0$ for all $t\in (\xi, \pi /2)$. So we have

Subcase $\bf{3.1}$: If $p\leq 32$, we have $G(0^{+})\leq G(\left(\pi /2\right) ^{-})$ and $\min \ \left(G(0^{+}), G(\left(\pi /2\right) ^{-})\right) = G(0^{+})$. The left-hand side inequality of $(3.3)$ leads to $(1.5)$.

Subcase $\bf{3.2}$: If $p > 32$, we have $G(\left(\pi /2\right) ^{-}) < G(0^{+})$ and $\min \ \left(G(0^{+}), G(\left(\pi /2\right) ^{-}) \right) = G(\left(\pi /2\right) ^{-})$. The left-hand side inequality of $(3.3)$ leads to $(1.6)$ by Lemma $6$.

So the proofs of Theorems 4–6 are complete.

Remark 1. Let us notice that proofs of all inequalities in new Theorems 4–6 also can be obtained forming appropriated mixed trigonometric polynomial function based on the function $g(t)$ by methods and algorithms developed in ^[18,19].

4.   Conclusions

We have established some sharp inequalities of Shafer-type for all $x\in (0, \infty)$:

The above inequalities improve and develop the known famous results.

Acknowledgments

The author is thankful to reviewers for careful corrections to and valuable comments on the original version of this paper. This paper is supported by the Natural Science Foundation of China grants No.61772025.

Conflict of interest

The author declares no conflict of interest in this paper.