# Some minimax approximants of D'Aurizio trigonometric inequalities

Branko Malesevic, Bojana Mihailović, Marija Nenezić Jović, Luka Milinković

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Branko Malesevic, Bojana Mihailović, Marija Nenezić Jović, Luka Milinković. Some minimax approximants of D'Aurizio trigonometric inequalities. 2022. hal-03550277

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Some generalisations and minimax approximants
of D'Aurizio trigonometric inequalities

May, 2024

Branko Malešević*, Bojana Mihailović,

Marija Nenezić Jović, Miloš Mićović, Luka Milinković

School of Electrical Engineering, University of Belgrade,
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

**Abstract.** In this paper, we generalise J. Sándor's results on D'Aurizio's trigonometric inequalities using stratified families of functions.

Keywords: D'Aurizio's trigonometric inequalities, stratified families of functions, a minimax approximant, Nike theorem

MSC 2020: Primary 41A44; Secondary 26D05

# 1 Introduction and Preliminaries

In this paper, we give some generalisations of the following results of József Sándor [1] concerning D'Aurizio's trigonometric inequalities:

**Theorem 1** (J. Sándor) For $0 < |x| < \pi/2$

$$ (1) \qquad 1 - \frac{4}{\pi^2} x^2 < \frac{\cos x}{\cos \frac{x}{2}} < 1 - \frac{3}{8} x^2 $$

holds.

**Theorem 2** (J. Sándor) For $0 < |x| < \pi/2$

$$ (2) \qquad 2 - \frac{1}{4} x^2 < \frac{\sin x}{\sin \frac{x}{2}} < 2 - \frac{4(2 - \sqrt{2})}{\pi^2} x^2 $$

holds.

<footnote>*corresponding author</footnote>
# Some minimax approximants of D'Aurizio trigonometric inequalities

February, 2022

Branko Malešević*, Bojana Mihailović,

Marija Nenezić Jović, Luka Milinković

School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia

**Abstract.** In this paper, we generalize J. Sándor's results on D'Aurizio's trigonometric inequalities using stratified families of functions.

Keywords: Inequalities, trigonometric functions

MSC 2020: Primary 33B10; Secondary 26D05

## 1 Introduction and Preliminaries

In this paper we give some generalizations of the following results of József Sándor [1] concerning D'Aurizio's trigonometric inequalities:

**Theorem 1** (J. Sándor) For $0 < |x| < \pi/2$

$$ (1) \qquad 1 - \frac{4}{\pi^2} x^2 < \frac{\cos x}{\cos \frac{x}{2}} < 1 - \frac{3}{8} x^2 $$

holds.

**Theorem 2** (J. Sándor) For $0 < |x| < \pi/2$

$$ (2) \qquad 2 - \frac{1}{4} x^2 < \frac{\sin x}{\sin \frac{x}{2}} < 2 - \frac{4(2 - \sqrt{2})}{\pi^2} x^2 $$

holds.

<footnote>*corresponding author</footnote>
The improved results are obtained using concepts presented in [2]. In this section, the important theorems from [2], which are necessary for further proofs, are listed.

Let

$$ \varphi_p(x) : (a, b) \to \mathbb{R} $$

be a family of functions with a variable $x \in (a, b)$ and a parameter $p \in \mathbb{R}^+$. In this paper, we call $\sup_{x \in (a,b)} |\varphi_p(x)|$ an error and denote it by:

$$ (3) \qquad d^{(p)} = \sup_{x \in (a,b)} |\varphi_p(x)|. $$

In [2], the conditions for the existence of the unique value $p_0$ of the parameter $p \in \mathbb{R}^+$, for which an infimum of an error is attained, are explored. Such infimum is denoted by:

$$ (4) \qquad d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (a,b)} |\varphi_p(x)|. $$

For such a value $p_0$, the function $\varphi_{p_0}(x)$ is called the minimax approximant on $(a, b)$. A family of functions $\varphi_p(x)$ is increasingly stratified if $p' > p'' \iff \varphi_{p'}(x) > \varphi_{p''}(x)$ for any $x \in (a, b)$ and, conversely, it is decreasingly stratified if $p' > p'' \iff \varphi_{p'}(x) < \varphi_{p''}(x)$ for any $x \in (a, b)$ ($p', p'' \in \mathbb{R}^+$).

Based on Theorem 1 and Theorem 1' from [2], we can conclude that for stratified families of functions, the following theorem is true:

**Theorem 3** Let $\varphi_p(x)$ be an increasingly (decreasingly) stratified family of functions (for $p \in \mathbb{R}^+$) that are continuous with respect to $x \in (a, b)$ for each $p \in \mathbb{R}^+$, and let $c, d$ be in $\mathbb{R}^+$ such that $c < d$. If:

(a) $\varphi_c(x) < 0$ ($\varphi_c(x) > 0$) and $\varphi_d(x) > 0$ ($\varphi_d(x) < 0$) for all $x \in (a, b)$, and at the endpoints $\varphi_c(a+) = \varphi_d(a+) = 0$, $\varphi_c(b-) = 0$ ($\varphi_d(b-) = 0$) and $\varphi_d(b-) \in \mathbb{R}^+$ ($\varphi_c(b-) \in \mathbb{R}^+$) hold;

(b) the functions $\varphi_p(x)$ are continuous with respect to $p \in (c, d)$ for each $x \in (a, b)$ and $\varphi_p(b-)$ is continuous with respect to $p \in (c, d)$ too;

(c) for all $p \in (c, d)$, there exists a right neighbourhood of point $a$ in which $\varphi_p(x) < 0$ holds and a left neighbourhood of point $b$ in which $\varphi_p(x) > 0$ holds;

(d) for all $p \in (c, d)$, the function $\varphi_p(x)$ has exactly one extremum $t^{(p)}$ on $(a, b)$, which is minimum;

then there exists exactly one solution $p_0$, for $p \in \mathbb{R}^+$, of the following equation

$$ |\varphi_p(t^{(p)})| = \varphi_p(b-) $$

and for $d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}(b-)$ we have

$$ d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (a,b)} |\varphi_p(x)|. $$

**Remark 4** Theorem 1 in [2] considers the case of an increasingly stratified family of functions, while Theorem 1' is analogous and considers the case of a decreasingly stratified family of functions. In this paper, both Theorems are unified in Theorem 3 and improved. Specifically, in condition (c), we have added that there exists a left neighbourhood of point $b$ in which $\varphi_p(x) > 0$ holds. Although the theorems in [2] were correct, this addition uniquely defines the function $\varphi_c(x)$ in Theorem 1 and the function $\varphi_d(x)$ in Theorem 1' from the paper [2].
The improved results are obtained due to concepts presented in [2]. In this section the important theorems from [2] which are necessary for further proofs are listed.

Let

$$ \varphi_p(x) : (a, b) \to \mathbb{R}, $$

be a family of functions with a variable $x \in (a, b)$ and a parameter $p \in \mathbb{R}^+$. In this paper, we call $\sup_{x \in (a,b)} |\varphi_p(x)|$ an error and denote it by:

$$ (3) \qquad d^{(p)} = \sup_{x \in (a,b)} |\varphi_p(x)|. $$

In [2] the conditions for existence of the unique value $p_0$ of the parameter $p \in \mathbb{R}^+$, for which an infimum of an error is attained, are explored. Such infimum is denoted by:

$$ (4) \qquad d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (a,b)} |\varphi_p(x)|. $$

For such a value $p_0$, the function $\varphi_{p_0}(x)$ is called the minimax approximant on $(a, b)$. A family of function $\varphi_p(x)$ is increasingly stratified if $p' > p'' \iff \varphi_{p'}(x) > \varphi_{p''}(x)$ for any $x \in (a, b)$ and, conversely, it is decreasingly stratified if $p' > p'' \iff \varphi_{p'}(x) < \varphi_{p''}(x)$ for any $x \in (a, b)$ ($p', p'' \in \mathbb{R}$).

Based on Theorem 1 and Theorem 1' from [2], we can conclude that for stratified families of functions the following theorem is true:

**Theorem 3** Let $\varphi_p(x)$ be a family of functions that are continuous with respect to $x \in (a, b)$ for each $p \in \mathbb{R}^+$ and increasingly (decreasingly) stratified for $p \in \mathbb{R}^+$, and let $c, d$ be in $\mathbb{R}^+$ such that $c < d$. If:

(a) $\varphi_c(x) < 0$ ($\varphi_c(x) > 0$) and $\varphi_d(x) > 0$ ($\varphi_d(x) < 0$) for all $x \in (a, b)$, and at the endpoints $\varphi_c(a+) = \varphi_d(a+) = 0$, $\varphi_c(b-) = 0$ ($\varphi_d(b-) = 0$) and $\varphi_d(b-) \in \mathbb{R}^+$ ($\varphi_c(b-) \in \mathbb{R}^+$) hold;

(b) the functions $\varphi_p(x)$ are continuous with respect to $p \in (c, d)$ for each $x \in (a, b)$ and $\varphi_p(b-)$ is continuous with respect to $p \in (c, d)$ too;

(c) for all $p \in (c, d)$, there exists a right neighbourhood of the point $a$ in which $\varphi_p(x) < 0$ holds;

(d) for all $p \in (c, d)$ the function $\varphi_p(x)$ has exactly one extremum $t^{(p)}$ on $(a, b)$, which is minimum;

then there exists exactly one solution $p_0$, for $p \in \mathbb{R}^+$, of the following equation

$$ |\varphi_p(t^{(p)})| = \varphi_p(b-) $$

and for $d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}(b-)$ we have

$$ d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (a,b)} |\varphi_p(x)|. $$

Exploring the fulfillment of the conditions for the Theorem 3 is often reduced to the following statement [2]:

**Theorem 4 (Nike theorem)** Let $f : (0, c) \to \mathbb{R}$ be $m$ times differentiable function (for some $m \ge 2, m \in \mathbb{N}$) satisfying the following conditions:
Exploring the fulfillment of the conditions for Theorem 3 is often reduced to the following statement [2]:

**Theorem 5 (Nike theorem)** Let $f: (0, c) \to \mathbb{R}$ be $m$ times differentiable function (for some $m \ge 2, m \in \mathbb{N}$) satisfying the following conditions:

(a) $f^{(m)}(x) > 0$ for $x \in (0, c)$;

(b) there is a right neighbourhood of zero in which the following inequalities are true:

$$f < 0, f' < 0, \dots, f^{(m-1)} < 0;$$

(c) there is a left neighbourhood of $c$ in which the following inequalities are true:

$$f > 0, f' > 0, \dots, f^{(m-1)} > 0.$$

Then the function $f$ has exactly one zero $x_0 \in (0, c)$, and $f(x) < 0$ for $x \in (0, x_0)$ and $f(x) > 0$ for $x \in (x_0, c)$. Also, the function $f$ has exactly one local minimum $t$ on the interval $(0, c)$. More precisely, there is exactly one point $t \in (0, c)$ (in fact $t \in (0, x_0)$) such that $f(t) < 0$ is the smallest value of the function $f$ on the interval $(0, c)$ and particularly on $(0, x_0)$.

# 2 Main results

In this section, some generalisations of Theorems 1 and 2 are given.

## Generalisation of Theorem 1

First, we give some auxiliary results.

**Lemma 1** The family of functions

$$\varphi_p(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p x^2 \quad (\text{for } x \in (0, \pi/2))$$

is decreasingly stratified with respect to parameter $p \in \mathbb{R}^+$.

The family of functions $\varphi_p(x)$, introduced in the previous lemma, is formed based on the double inequality from Theorem 1 for parameter values $p = \frac{4}{\pi^2}$ and $p = \frac{3}{8}$, as will be discussed in the following analysis. With that aim, we introduce the function

$$g(x) = \frac{-2 \cos^2 \frac{x}{2} + \cos \frac{x}{2} + 1}{x^2 \cos \frac{x}{2}} \quad (\text{for } x \in (0, \pi/2))$$

which is strictly increasing, while $g(0+) = \frac{3}{8}$ and $g(\pi/2) = \frac{4}{\pi^2}$ hold [1]. Obviously,

$$\varphi_p(x) = 0 \Leftrightarrow p = g(x).$$
(a) $f^{(m)}(x) > 0$ for $x \in (0, c)$;

(b) there is a right neighbourhood of zero in which the following inequalities are true:

$$f < 0, f' < 0, \dots, f^{(m-1)} < 0;$$

(c) there is a left neighbourhood of c in which the following inequalities are true:

$$f > 0, f' > 0, \dots, f^{(m-1)} > 0.$$

Then the function $f$ has exactly one zero $x_0 \in (0, c)$, and $f(x) < 0$ for $x \in (0, x_0)$ and $f(x) > 0$ for $x \in (x_0, c)$. Also, the function $f$ has exactly one local minimum $t$ on the interval $(0, c)$. More precisely, there is exactly one point $t \in (0, c)$ (in fact $t \in (0, x_0)$) such that $f(t) < 0$ is the smallest value of the function $f$ on the interval $(0, c)$ and particularly on $(0, x_0)$.

# 2 Main results

In this section some generalizations of Theorems 1 and 2 are shown.

## Generalization of the Theorem 1

First we give some auxiliary results.

**Lemma 1** The family of functions

$$\varphi_p(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p x^2 \quad (\text{for } x \in (0, \pi/2))$$

is decreasingly stratified with respect to parameter $p \in \mathbb{R}^+$.

Let us introduce the function

$$g(x) = \frac{-2 \cos^2 \frac{x}{2} + \cos \frac{x}{2} + 1}{x^2 \cos \frac{x}{2}} \quad (\text{for } x \in (0, \pi/2))$$

which is strictly increasing, while $g(0+) = \frac{3}{8}$ and $g(\pi/2) = \frac{4}{\pi^2}$ hold [1]. Obviously,

$$\varphi_p(x) = 0 \Leftrightarrow p = g(x).$$

Now we give the main result for the first generalisation:

**Statement 1** Let

$$A = \frac{3}{8} = 0.375 \quad \text{and} \quad B = \frac{4}{\pi^2} = 0.40528 \dots$$

(i) If $p \in (0, A]$ then

$$x \in (0, \pi/2) \implies \frac{\cos x}{\cos \frac{x}{2}} < 1 - A x^2 < 1 - p x^2.$$
Now we give the main result for the first generalisation:

**Statement 1** Let

$$A = \frac{3}{8} = 0.375 \quad \text{and} \quad B = \frac{4}{\pi^2} = 0.40528 \dots$$

(i) If $p \in (0, A]$, then

$$x \in \left(0, \frac{\pi}{2}\right) \implies \frac{\cos x}{\cos \frac{x}{2}} < 1 - A x^2 < 1 - p x^2.$$

(ii) If $p \in (A, B)$, then $\varphi_p(x)$ has exactly one zero $x_0^{(p)}$ on $(0, \pi/2)$. Also,

$$x \in \left(0, x_0^{(p)}\right) \implies \frac{\cos x}{\cos \frac{x}{2}} > 1 - p x^2$$

and

$$x \in \left(x_0^{(p)}, \frac{\pi}{2}\right) \implies \frac{\cos x}{\cos \frac{x}{2}} < 1 - p x^2$$

hold.

(iii) If $p \in [B, \infty)$, then

$$x \in \left(0, \frac{\pi}{2}\right) \implies \frac{\cos x}{\cos \frac{x}{2}} > 1 - B x^2 > 1 - p x^2.$$

**Proof.** The function $g(x)$ is increasing, continuous and surjection on $(A, B)$, see [1]. It is obvious that

$$g(x) - p = \frac{\varphi_p(x)}{x^2}$$

holds. Therefore, $g(x) \neq p$ (i.e. $\varphi_p(x) \neq 0$) holds on $(0, \frac{\pi}{2})$ if $p \in (0, A]$ or $p \in [B, +\infty)$. We can easily see that $\varphi_A(\pi/2) > 0$ and $\varphi_B(\pi/3) < 0$. Hence, $\varphi_A(x) > 0$ for $x \in (0, \frac{\pi}{2})$ and $\varphi_B(x) < 0$ for $x \in (0, \frac{\pi}{2})$. Then (i) and (iii) follow from the decreasing stratification of the family $\varphi_p(x)$. Furthermore, for $p \in (A, B)$, the equation $g(x) = p$ has exactly one solution, which we denote by $x_0^{(p)}$, while $g(x) < p$ for $x \in (0, x_0^{(p)})$ and $g(x) > p$ for $x \in (x_0^{(p)}, \frac{\pi}{2})$. Hence, (ii) is true. $\square$

**Corollary 1** For any $0 < x < \pi/2$

$$1 - \frac{4}{\pi^2} x^2 < \frac{\cos x}{\cos \frac{x}{2}} < 1 - \frac{3}{8} x^2$$

holds, with the best possible constants $A = \frac{3}{8} = 0.375$ and $B = \frac{4}{\pi^2} = 0.40528 \dots$.

**Statement 2** Let

$$\varphi_p(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p x^2 \text{ for } x \in (0, \frac{\pi}{2}) \text{ and } p \in \mathbb{R}^+.$$

(i) For $p \in (A, B)$, there exists only one extremum of this function on $(0, \frac{\pi}{2})$ at $t^{(p)}$ and that extremum is minimum.
(ii) If $p \in (A, B)$, then $\varphi_p(x)$ has exactly one zero $x_0^{(p)}$ on $(0, \pi/2)$. Also,

$$x \in (0, x_0^{(p)}) \implies \frac{\cos x}{\cos \frac{x}{2}} > 1 - p x^2$$

and

$$x \in (x_0^{(p)}, \pi/2) \implies \frac{\cos x}{\cos \frac{x}{2}} < 1 - p x^2$$

hold.

(iii) If $p \in [B, \infty)$ then

$$x \in (0, \pi/2) \implies \frac{\cos x}{\cos \frac{x}{2}} > 1 - B x^2 > 1 - p x^2.$$

**Proof.** The function $g(x)$ is increasing, continuous and onto $(A, B)$, see [1]. It is obvious that

$$g(x) - p = \frac{\varphi_p(x)}{x^2}$$

holds. Therefore, $g(x) \neq p$ (e.g. $\varphi_p(x) \neq 0$) holds on $(0, \frac{\pi}{2})$ if $p \in (0, A]$ or $p \in [B, +\infty)$. We can easily spot that $\varphi_A(\pi/2) > 0$ and $\varphi_B(\pi/3) < 0$. Hence, $\varphi_A(x) > 0$ for $x \in (0, \frac{\pi}{2})$ and $\varphi_B(x) < 0$ for $x \in (0, \frac{\pi}{2})$. Then (i) and (iii) follow from the decreasingly stratification of the family $\varphi_p(x)$. Furthermore, for $p \in (A, B)$ the equation $g(x) = p$ has exactly one solution, which we denote by $x_0^{(p)}$, while $g(x) < p$ for $x \in (0, x_0^{(p)})$ and $g(x) > p$ for $x \in (x_0^{(p)}, \frac{\pi}{2})$. Hence, (ii) is true. $\square$

**Corollary 1** For any $0 < x < \pi/2$

$$1 - \frac{4}{\pi^2} x^2 < \frac{\cos x}{\cos \frac{x}{2}} < 1 - \frac{3}{8} x^2$$

holds, with the best possible constants $A = \frac{3}{8} = 0.375$ and $B = \frac{4}{\pi^2} = 0.40528...$.

**Statement 2** Let

$$\varphi_p(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p x^2 \text{ for } x \in (0, \frac{\pi}{2}) \text{ and } p \in \mathbb{R}^+.$$

(i) For $p \in (A, B)$ there exists only one extremum of this function on $(0, \frac{\pi}{2})$ in $t^{(p)}$ and that extremum is minimum.

(ii) There is exactly one solution of the equation

$$|\varphi_p(t^{(p)})| = \varphi_p(\frac{\pi}{2}) -$$

with the respect to parameter $p \in (A, B)$, which can be determined numerically as

$$p_0 = 0.39916...$$

For the value

$$d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}(\frac{\pi}{2}) = 0.015109...$$
(ii) There is exactly one solution to the equation

$$|\varphi_p(t^{(p)})| = \varphi_p\left(\frac{\pi}{2}\right)$$

with the respect to parameter $p \in (A, B)$, which can be determined numerically as

$$p_0 = 0.39916...$$

For the value

$$d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}\left(\frac{\pi}{2}\right) = 0.015109...,$$

the following result

$$d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (0, \pi/2)} |\varphi_p(x)|$$

holds.

(iii) The minimax approximant of the family $\varphi_p(x)$ is

$$\varphi_{p_0}(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p_0 x^2,$$

which determines the corresponding minimax approximation

$$\frac{\cos x}{\cos \frac{x}{2}} \approx 1 - 0.39916 x^2.$$

**Proof.** For $p \in (A, B)$, functions $\varphi_p(x)$ fulfill the conditions of Theorem 5 (Nike theorem):

(a) For $m = 3$

$$\varphi_p''(x) = \frac{d^3 \varphi_p}{dx^3} = \frac{1}{8} \frac{\left(6 - 2 \cos^4 \frac{x}{2} - \cos^2 \frac{x}{2}\right) \sin \frac{x}{2}}{\cos^4 \frac{x}{2}} > 0 \quad (x \in (0, \pi/2)).$$

(b) Based on the Taylor expansions of the functions $\varphi_p(x)$ around $x=0$:

$$(5) \qquad \varphi_p(x) = \left(\frac{3}{8} - p\right) x^2 + \frac{1}{128} x^4 + o(x^4),$$

there exists a right neighbourhood $\mathcal{U}_0$ of the point 0 such that

$$\varphi_p(x), \varphi'_p(x) = \frac{d\varphi_p}{dx}, \varphi''_p(x) = \frac{d^2\varphi_p}{dx^2} < 0 \quad (x \in \mathcal{U}_0).$$

(c) Based on the Taylor expansions of the functions $\varphi_p(x)$ around $x = \frac{\pi}{2}$:

$$(6) \qquad \varphi_p(x) = \left(1 - \frac{p\pi^2}{4}\right) + \left(-p\pi + \sqrt{2}\right)\left(x - \frac{\pi}{2}\right) + \\ \qquad \qquad + \left(\frac{\sqrt{2}}{2} - p\right)\left(x - \frac{\pi}{2}\right)^2 + \frac{5\sqrt{2}}{24}\left(x - \frac{\pi}{2}\right)^3 + o\left(\left(x - \frac{\pi}{2}\right)^3\right),$$

there exists a left neighbourhood $\mathcal{U}_{\pi/2}$ of the point $\pi/2$ such that

$$\varphi_p(x), \varphi'_p(x) = \frac{d\varphi_p}{dx}, \varphi''_p(x) = \frac{d^2\varphi_p}{dx^2} > 0 \quad (x \in \mathcal{U}_{\pi/2}).$$
the following result

$$d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (0, \pi/2)} |\varphi_p(x)|$$

holds.

(iii) The minimax approximant of the family $\varphi_p(x)$ is

$$\varphi_{p_0}(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p_0 x^2$$

which determines the appropriate minimax approximation

$$\frac{\cos x}{\cos \frac{x}{2}} \approx 1 - 0.39916 x^2.$$

**Proof.** For $p \in (A, B)$ functions $\varphi_p(x)$ fulfill the conditions of the Theorem 4 (Nike theorem):

(a) For $m = 3$

$$\varphi_p'''(x) = \frac{d^3 \varphi_p}{dx^3} = \frac{1}{8} \frac{\left(6 - 2 \cos^4 \frac{x}{2} - \cos^2 \frac{x}{2}\right) \sin \frac{x}{2}}{\cos^4 \frac{x}{2}} > 0 \quad (x \in (0, \pi/2)).$$

(b) Based on Taylor expansions of the functions $\varphi_p(x)$ around $x=0$

$$(5) \qquad \varphi_p(x) = \left(\frac{3}{8} - p\right) x^2 + \frac{1}{128} x^4 + o(x^4),$$

there exists a right neighbourhood $\mathcal{U}_0$ of the point 0 such that

$$\varphi_p(x), \varphi'_p(x) = \frac{d\varphi_p}{dx}, \varphi''_p(x) = \frac{d^2\varphi_p}{dx^2} < 0 \quad (x \in \mathcal{U}_0).$$

(c) Based on Taylor expansions of the functions $\varphi_p(x)$ around $x=\frac{\pi}{2}$

$$(6) \qquad \varphi_p(x) = \left(1 - \frac{p\pi^2}{4}\right) + \left(-p\pi + \sqrt{2}\right)\left(x - \frac{\pi}{2}\right) + \\ \qquad \qquad + \left(\frac{\sqrt{2}}{2} - p\right)\left(x - \frac{\pi}{2}\right)^2 + \frac{5\sqrt{2}}{24}\left(x - \frac{\pi}{2}\right)^3 + o\left(\left(x - \frac{\pi}{2}\right)^3\right),$$

there exists a left neighbourhood $\mathcal{U}_{\pi/2}$ of the point $\pi/2$ such that

$$\varphi_p(x), \varphi'_p(x) = \frac{d\varphi_p}{dx}, \varphi''_p(x) = \frac{d^2\varphi_p}{dx^2} > 0 \quad (x \in \mathcal{U}_{\pi/2}).$$

Due to Theorem 3, for $p \in (A, B)$ we can conclude that functions $\varphi_p(x)$ has exactly one extreme $t^{(p)}$, which is minimum, on $(0, \frac{\pi}{2})$ (and, of course, exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$).

The family of functions $\varphi_p(x)$, for values $p \in (A, B)$, fulfills the conditions of Theorem 3, hence there exists a minimax approximant. Numerical determination of the
Based on Theorem 3, for $p \in (A, B)$, we can conclude that each function $\varphi_p(x)$ has exactly one extremum $t^{(p)}$, which is minimum, on $(0, \frac{\pi}{2})$ (and thus exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$).

The family of functions $\varphi_p(x)$, for values $p \in (A, B)$, fulfills the conditions of Theorem 3, thereby there exists a minimax approximant. Numerical determination of the minimax approximant and the error can be calculated in Maple in the manner we present here. Let $f(x, p) := \varphi_p(x)$ and $F(x, p) := \varphi'_p(x)$. With Maple code

$$ \texttt{fsolve}(\{F(x,p) = 0, \text{abs}(f(x,p)) = f(\pi/2, p)\}, \{x = 0..\pi/2, p = A..B\}); $$

we have numerical values

$$ \{p = 0.399161163, x = 1.069252853\}. $$

For the value $p_0 = 0.39916...$, we have the minimax approximant of the family

$$ \varphi_{p_0}(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p_0 x^2 $$

and numerical value of minimax error

$$ d_0 = f(\pi/2, p_0) = 0.015109... \quad \square $$

Figure 1 illustrates the stratified family of functions from Lemma 1 for $p \in \mathbb{R}^+$.

<figure><img src="image_1.png" /><figcaption>Figure 1: Stratified family of functions from Lemma 1</figcaption></figure>
minimax approximant and the error can be calculated in Maple in the way we present here. Let $f(x,p) := \varphi_p(x)$ and $F(x,p) := \varphi'_p(x)$. With Maple code

$$fsolve(\{F(x,p) = 0, \text{abs}(f(x,p)) = f(\pi/2,p)\}, \{x = 0..\pi/2, p = A..B\});$$

we have numerical values

$$\{p = 0.399161163, x = 1.069252853\}$$

For the value $p_0 = 0.39916...$ we have the minimax approximant of the family

$$\varphi_{p_0}(x) = 1 - \frac{\cos x}{\cos \frac{x}{2}} - p_0 x^2$$

and numerical value of minimax error

$$d_0 = f(\pi/2, p_0) = 0.015109... \quad \square$$

**Generalization of the Theorem 2**

**Lemma 2** Family of functions

$$\varphi_p(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p x^2 \quad (\text{for } x \in (0, \pi/2))$$

is increasingly stratified with respect to parameter $p \in \mathbb{R}^+$.

Let us notice that the function

$$g(x) = \frac{2\left(1 - \cos\frac{x}{2}\right)}{x^2} \quad (\text{for } x \in (0, \pi/2))$$

is strictly decreasing and that $\lim_{x \to 0^+} g(x) = \frac{1}{4}$ and $\lim_{x \to \pi/2^-} g(x) = \frac{8-4\sqrt{2}}{\pi^2}$. Further,

$$\varphi_p(x) = 0 \Leftrightarrow p = g(x).$$

holds, as in the previous case.

Now we give the main result for the second generalisation:

**Statement 3** Let

$$A = \frac{8 - 4\sqrt{2}}{\pi^2} = 0.23741... \quad \text{and} \quad B = \frac{1}{4} = 0.25.$$

(i) If $p \in (0, A]$ then

$$x \in (0, \pi/2) \implies \frac{\sin x}{\sin \frac{x}{2}} < 2 - A x^2 < 2 - p x^2.$$

(ii) If $p \in (A, B)$, then $\varphi_p(x)$ has exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$. Also,

$$x \in (0, x_0^{(p)}) \implies \frac{\sin x}{\sin \frac{x}{2}} < 2 - p x^2$$

and

$$x \in (x_0^{(p)}, \pi/2) \implies \frac{\sin x}{\sin \frac{x}{2}} > 2 - p x^2$$

hold.
**Generalisation of Theorem 2**

First, we give some auxiliary results.

**Lemma 2** The family of functions

$$ \varphi_p(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p x^2 \quad (\text{for } x \in (0, \pi/2)) $$

is increasingly stratified with respect to parameter $p \in \mathbb{R}^+$.

The family of functions $\varphi_p(x)$, introduced in the previous lemma, is formed based on the double inequality from Theorem 2 for parameter values $p = \frac{1}{4}$ and $p = \frac{8-4\sqrt{2}}{\pi^2}$, as will be discussed in the following analysis. With that aim, we introduce the function

$$ g(x) = \frac{2 \left(1 - \cos \frac{x}{2}\right)}{x^2} \quad (\text{for } x \in (0, \pi/2)) $$

which is strictly decreasing, while $g(0+) = \frac{1}{4}$ and $g(\pi/2-) = \frac{8-4\sqrt{2}}{\pi^2}$ hold. Further,

$$ \varphi_p(x) = 0 \Leftrightarrow p = g(x). $$

holds, as in the previous case.

Now we give the main result for the second generalisation:

**Statement 3** Let

$$ A = \frac{8 - 4\sqrt{2}}{\pi^2} = 0.23741 \dots \quad \text{and} \quad B = \frac{1}{4} = 0.25. $$

(i) If $p \in (0, A]$, then

$$ x \in (0, \frac{\pi}{2}) \implies \frac{\sin x}{\sin \frac{x}{2}} < 2 - A x^2 < 2 - p x^2. $$

(ii) If $p \in (A, B)$, then $\varphi_p(x)$ has exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$. Also,

$$ x \in (0, x_0^{(p)}) \implies \frac{\sin x}{\sin \frac{x}{2}} < 2 - p x^2 $$

and

$$ x \in (x_0^{(p)}, \frac{\pi}{2}) \implies \frac{\sin x}{\sin \frac{x}{2}} > 2 - p x^2 $$

hold.

(iii) If $p \in [B, \infty)$, then

$$ x \in (0, \frac{\pi}{2}) \implies \frac{\sin x}{\sin \frac{x}{2}} > 2 - B x^2 > 2 - p x^2. $$
(iii) If $p \in [B, \infty)$ then

$$x \in (0, \pi/2) \implies \frac{\sin x}{\sin \frac{\pi}{2}} > 2 - Bx^2 > 2 - px^2.$$

**Proof.** The function $g(x)$ is decreasing, continuous and onto $(A, B)$ and

$$g(x) - p = - \frac{\varphi_p(x)}{x^2}$$

holds. Hence, $\varphi_p(x) \neq 0$ holds on $(0, \frac{\pi}{2})$ if $p \in (0, A]$ or $p \in [B, +\infty)$. It can be checked that $\varphi_A(\pi/3) < 0$ and $\varphi_B(\pi/2) > 0$, which means that $\varphi_A(x) < 0$ for $x \in (0, \frac{\pi}{2})$ and $\varphi_B(x) > 0$ for $x \in (0, \frac{\pi}{2})$. Then (i) and (iii) follow from the increasingly stratification of the family $\varphi_p(x)$. For $p \in (A, B)$, the equation $g(x) = p$ has exactly one solution which we denote by $x_0^{(p)}$, while $g(x) > p$ for $x \in (0, x_0^{(p)})$ and $g(x) < p$ for $x \in (x_0^{(p)}, \frac{\pi}{2})$. Hence, (ii) holds. $\square$

**Corollary 2** For any $0 < x < \pi/2$

$$2 - \frac{1}{4}x^2 < \frac{\sin x}{\sin \frac{\pi}{2}} < 2 - \frac{4(2 - \sqrt{2})}{\pi^2}x^2.$$

holds, with the best possible constants $A = \frac{8 - 4\sqrt{2}}{\pi^2} = 0.23741...$ and $B = \frac{1}{4} = 0.25$.

**Statement 4** Let

$$\varphi_p(x) = -2 + \frac{\sin x}{\sin \frac{\pi}{2}} + px^2 \text{ for } x \in (0, \frac{\pi}{2}) \text{ and } p \in \mathbb{R}^+.$$

(i) For $p \in (A, B)$ there exists only one extremum of this function on $(0, \frac{\pi}{2})$ in $t^{(p)}$ and that extremum is minimum.

(ii) There is exactly one solution of the equation

$$|\varphi_p(t^{(p)})| = \varphi_p(\frac{\pi}{2} - ),$$

where $t^{(p)}$ is unique local minimum of $\varphi_p(x)$, by parameter $p \in (A, B)$ which we determine numerically as

$$p_0 = 0.23955...$$

For the value

$$d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}(\frac{\pi}{2} - ) = 0.0052842...$$

the following result

$$d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (0, \pi/2)} |\varphi_p(x)|.$$

holds.

(iii) The minimax approximant of the family $\varphi_p(x)$ is

$$\varphi_{p_0}(x) = -2 + \frac{\sin x}{\sin \frac{\pi}{2}} + p_0 x^2$$

which determines the appropriate minimax approximation

$$\frac{\sin x}{\sin \frac{\pi}{2}} \approx 2 - 0.23955 x^2.$$
**Proof.** The function $g(x)$ is increasing, continuous and surjection on $(A, B)$, and

$$g(x) - p = - \frac{\varphi_p(x)}{x^2}$$

holds. Hence, $\varphi_p(x) \neq 0$ holds on $(0, \frac{\pi}{2})$ if $p \in (0, A]$ or $p \in [B, +\infty)$. It can be checked that $\varphi_A(\pi/3) < 0$ and $\varphi_B(\pi/2) > 0$, which means that $\varphi_A(x) < 0$ for $x \in (0, \frac{\pi}{2})$ and $\varphi_B(x) > 0$ for $x \in (0, \frac{\pi}{2})$. Then (i) and (iii) follow from the increasing stratification of the family $\varphi_p(x)$. For $p \in (A, B)$, the equation $g(x) = p$ has exactly one solution which we denote by $x_0^{(p)}$, while $g(x) > p$ for $x \in (0, x_0^{(p)})$ and $g(x) < p$ for $x \in (x_0^{(p)}, \frac{\pi}{2})$. Hence, (ii) holds. $\square$

**Corollary 2** For any $0 < x < \pi/2$

$$2 - \frac{1}{4}x^2 < \frac{\sin x}{\sin \frac{x}{2}} < 2 - \frac{4(2 - \sqrt{2})}{\pi^2}x^2.$$

holds, with the best possible constants $A = \frac{8 - 4\sqrt{2}}{\pi^2} = 0.23741...$ and $B = \frac{1}{4} = 0.25$.

**Statement 4** Let

$$\varphi_p(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p x^2 \text{ for } x \in (0, \frac{\pi}{2}) \text{ and } p \in \mathbb{R}^+.$$

(i) For $p \in (A, B)$, there exists only one extremum of this function on $(0, \frac{\pi}{2})$ at $t^{(p)}$ and that extremum is minimum.

(ii) There is exactly one solution to the equation

$$|\varphi_p(t^{(p)})| = \varphi_p(\frac{\pi}{2}-),$$

where $t^{(p)}$ is a unique local minimum of $\varphi_p(x)$, by parameter $p \in (A, B)$, which we determine numerically as

$$p_0 = 0.23955...$$

For the value

$$d_0 = |\varphi_{p_0}(t^{(p_0)})| = \varphi_{p_0}(\frac{\pi}{2}-) = 0.0052842...,$$

the following result

$$d_0 = \inf_{p \in \mathbb{R}^+} \sup_{x \in (0, \pi/2)} |\varphi_p(x)|$$

holds.

(iii) The minimax approximant of the family $\varphi_p(x)$ is

$$\varphi_{p_0}(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p_0 x^2,$$

which determines the corresponding minimax approximation

$$\frac{\sin x}{\sin \frac{x}{2}} \approx 2 - 0.23955 x^2.$$
**Proof.** For $p \in (A, B)$ functions $\varphi_p(x)$ fulfill the conditions of the Theorem 4 (Nike theorem):

(a) For $m = 3$

$$ \varphi_p'''(x) = \frac{d^3\varphi_p}{dx^3} = \frac{1}{4}\sin\frac{x}{2} > 0 \quad (x \in (0, \pi/2)). $$

(b) Based on Taylor's expansion of the function $\varphi_p(x)$ around $x=0$

$$ (7) \qquad \varphi_p(x) = \left(-\frac{1}{4} + p\right)x^2 + \frac{1}{192}x^4 + o(x^4) $$

there is a right neighbourhood $U_0$ of the point 0 such that

$$ \varphi_p(x), \varphi_p'(x) = \frac{d\varphi_p}{dx}, \varphi_p''(x) = \frac{d^2\varphi_p}{dx^2} < 0 \quad (x \in U_0). $$

(c) Based on Taylor's expansion of the function $\varphi_p(x)$ around $x=\frac{\pi}{2}$

$$ (8) \qquad \varphi_p(x) = \left(-2 + \sqrt{2} + \frac{p\pi^2}{4}\right) + \left(p\pi - \frac{\sqrt{2}}{2}\right)\left(x - \frac{\pi}{2}\right) + \\ \qquad \qquad + \left(p - \frac{\sqrt{2}}{8}\right)\left(x - \frac{\pi}{2}\right)^2 + \frac{\sqrt{2}}{48}\left(x - \frac{\pi}{2}\right)^3 + o\left(\left(x - \frac{\pi}{2}\right)^3\right) $$

there is left neighbourhood $U_{\pi/2}$ of the point $\pi/2$ such that it is

$$ \varphi_p(x), \varphi_p'(x) = \frac{d\varphi_p}{dx}, \varphi_p''(x) = \frac{d^2\varphi_p}{dx^2} > 0 \quad (x \in U_{\pi/2}). $$

Due to Theorem 3, for $p \in (A, B)$ we can conclude that functions $\varphi_p(x)$ has exactly one extreme $t^{(p)}$, which is minimum, on $(0, \frac{\pi}{2})$ (and, of course, exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$).

The family of functions $\varphi_p(x)$, for values $p \in (A, B)$, fulfills the conditions of Theorem 3, hence there exists a minimax approximant. Numerical determination of the minimax approximant and the error can be calculated in Maple in the way we present here. Let $f(x, p) := \varphi_p(x)$ and $F(x, p) := \varphi'_p(x)$. With Maple code

$$ \texttt{fsolve}(\{F(x,p) = 0, \texttt{abs}(f(x,p)) = f(\pi/2,p)\}, \{x = 0..\pi/2, p = A..B\}); $$

we have numerical values

$$ \{p = 0.2395519170, x = 1.007887451\} $$

For the value $p_0 = 0.23955...$ we have the minimax approximant of the family

$$ \varphi_{p_0}(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p_0 x^2 $$

and numerical value of minimax error

$$ d_0 = f(\pi/2, p_0) = 0.0052842... \quad \square $$
**Proof.** For $p \in (A, B)$, functions $\varphi_p(x)$ fulfill the conditions of Theorem 5 (Nike theorem):

(a) For $m = 3$

$$ \varphi_p'''(x) = \frac{d^3\varphi_p}{dx^3} = \frac{1}{4}\sin\frac{x}{2} > 0 \quad (x \in (0, \pi/2)). $$

(b) Based on the Taylor expansion of the functions $\varphi_p(x)$ around $x=0$:

$$ (7) \qquad \varphi_p(x) = \left(-\frac{1}{4} + p\right)x^2 + \frac{1}{192}x^4 + o(x^4) $$

there exists a right neighbourhood $U_0$ of the point 0 such that

$$ \varphi_p(x), \varphi_p'(x) = \frac{d\varphi_p}{dx}, \varphi_p''(x) = \frac{d^2\varphi_p}{dx^2} < 0 \quad (x \in U_0). $$

(c) Based on the Taylor expansion of the functions $\varphi_p(x)$ around $x = \frac{\pi}{2}$:

$$ (8) \qquad \varphi_p(x) = \left(-2 + \sqrt{2} + \frac{p\pi^2}{4}\right) + \left(p\pi - \frac{\sqrt{2}}{2}\right)\left(x - \frac{\pi}{2}\right) + \\ \qquad \qquad \qquad + \left(p - \frac{\sqrt{2}}{8}\right)\left(x - \frac{\pi}{2}\right)^2 + \frac{\sqrt{2}}{48}\left(x - \frac{\pi}{2}\right)^3 + o\left(\left(x - \frac{\pi}{2}\right)^3\right) $$

there exists a left neighbourhood $U_{\pi/2}$ of the point $\pi/2$ such that it is

$$ \varphi_p(x), \varphi_p'(x) = \frac{d\varphi_p}{dx}, \varphi_p''(x) = \frac{d^2\varphi_p}{dx^2} > 0 \quad (x \in U_{\pi/2}). $$

Based on Theorem 3, for $p \in (A, B)$, we can conclude that functions $\varphi_p(x)$ has exactly one extremum $t^{(p)}$, which is minimum, on $(0, \frac{\pi}{2})$ (and thus exactly one zero $x_0^{(p)}$ on $(0, \frac{\pi}{2})$).

The family of functions $\varphi_p(x)$, for values $p \in (A, B)$, fulfills the conditions of Theorem 3, thereby there exists a minimax approximant. Numerical determination of the minimax approximant and the error can be calculated in Maple in the manner we present here. Let $f(x, p) := \varphi_p(x)$ and $F(x, p) := \varphi'_p(x)$. With Maple code

$$ \texttt{fsolve}(\{F(x,p) = 0, \text{abs}(f(x,p)) = f(\pi/2,p)\}, \{x = 0..\pi/2, p = A..B\}); $$

we have numerical values

$$ \{p = 0.2395519170, x = 1.007887451\}. $$

For the value $p_0 = 0.23955...$, we have the minimax approximant of the family

$$ \varphi_{p_0}(x) = -2 + \frac{\sin x}{\sin \frac{x}{2}} + p_0 x^2 $$

and numerical value of minimax error

$$ d_0 = f(\pi/2, p_0) = 0.0052842... \quad \square $$

Figure 2 illustrates the stratified family of functions from Lemma 2 for $p \in \mathbb{R}^+$.
# 3 Conclusion

This paper specifies the results of J. Sándor [1] related to D'Aurizio's trigonometric inequality [8] using concepts from the paper [2]. Let us emphasize that in the paper [2] is presented one method for possible improvements of existing results in Theory of Analytic inequalities in terms of determining the corresponding minimax approximants for a number of inequalities from reviewed papers [6], [7], and books [3], [4], [5] as well. In further papers the subject of our studies will be to determine the appropriate minimax approximants for papers [9] and [10] relating to the generalizations of D'Aurizio's trigonometric inequalities.

## Acknowledgment

The authors are grateful to anonymous referees for their careful corrections to and valuable comments on the original version of this paper. Research of the first author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under Projects ON 174032 and III 44006 - continuation.

# References

[1] J. SÁNDOR: *On D'Aurizio's trigonometric inequality*, J. Math. Inequal. **10**:3 (2016), 885-888.

[2] B. MALEŠEVIĆ, B. MIHALOVIĆ: *A Minimax Approximant in the Theory of Analytical Inequalities*, Appl. Anal. Discrete Math. **15** (2021), 486-509.

[3] D. S. MITRINOVIĆ: *Analytic Inequalities*, Springer-Verlag, 1970.

[4] G. MILOVANOVIĆ, M. RASSIAS (editors), *Analytic Number Theory, Approximation Theory and Special Functions*, Springer 2014, Chapter: G. D. ANDERSON, M. VUORINEN, X. ZHANG: *Topics in Special Functions III*, 297-345.

[5] M.J. CLOUD, B.C. DRACHMAN, L.P. LEBEDEV: *Inequalities with Applications to Engineering*, Springer 2014.

[6] L. ZHU: *A source of inequalities for circular functions*, Comput. Math. Appl. **58** (2009), 1998-2004.

[7] F. QI, D.-W. NIU, B.-N. GUO *Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems*, J. Inequal. Appl. (Article ID 271923) (2009), 1-52 pages

[8] J. D'AURIZIO: *Refinements of the Shafer-Fink inequality of arbitrary uniform precision*, Math. Inequal. Appl. **17**:4 (2014), 1487-1498.

[9] J. SÁNDOR: *Extensions of D'Aurizio's trigonometric inequality*, Notes Number Theory Discrete Math. **23**:2 (2017), 81-83.

[10] L.-C. HUNG, P.-Y. LI: *On generalization of D'Aurizio-Sándor inequalities involving a parameter*, J. Math. Inequal. **12**:3 (2018), 853-860.
<figure><img src="image_2.png" /><figcaption>Figure 2: Stratified family of functions from Lemma 2</figcaption></figure>

## 3 Conclusion

This paper specifies the results of J. Sándor [1] related to D'Aurizio's trigonometric inequality [8] using concepts from the paper [2]. Additionally, Theorems 1 and 1' from [2] were improved. Let us emphasize that the paper [2] presents one method for possible improvements of existing results in the Theory of analytic inequalities in terms of determining the corresponding minimax approximants for many inequalities from the papers [6], [7], and the books [3]-[5]. The concept of stratification is used in recent research to improve and generalise some inequalities, see the papers [11]-[14], and can be used to improve many more from [3]-[5], [10], [15]-[21]. In further papers, the subject of our studies will be to determine the appropriate minimax approximants for papers [9] and [10] relating to the generalizations of D'Aurizio's trigonometric inequalities.

## Acknowledgment

This work was financially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia under contract numbers: 451-03-65/2024-03/200103 (for the first and second authors) and 451-03-66/2024-03/200103 (for the fourth author).

## References

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[2] B. MALEŠEVIĆ, B. MIHALOVIĆ: *A Minimax Approximant in the Theory of Analytical Inequalities*, Appl. Anal. Discrete Math. **15** (2021), 486–509.

[3] D. S. MITRINOVIĆ: *Analytic Inequalities*, Springer-Verlag, 1970.
[4] G. MILOVANOVNIĆ, M. RASSIAS (editors), *Analytic Number Theory, Approximation Theory and Special Functions*, Springer 2014, Chapter: G. D. ANDERSON, M. VUORINEN, X. ZHANG: *Topics in Special Functions III*, 297-345.

[5] M.J. CLOUD, B.C. DRACHMAN, L.P. LEBEDEV: *Inequalities with Applications to Engineering*, Springer 2014.

[6] L. ZHU: *A source of inequalities for circular functions*, Comput. Math. Appl. **58** (2009), 1998-2004.

[7] F. QI, D.-W. NIU, B.-N. GUO *Refinements, Generalizations, and Applications of Jordan's Inequality and Related Problems*, J. Inequal. Appl. (Article ID 271923) (2009), 1-52 pages

[8] J. D'AURIZIO: *Refinements of the Shafer-Fink inequality of arbitrary uniform precision*, Math. Inequal. Appl. **17**:4 (2014), 1487-1498.

[9] J. SÁNDOR: *Extensions of D'Aurizio's trigonometric inequality*, Notes Number Theory Discrete Math. **23**:2 (2017), 81-83.

[10] L.-C. HUNG, P.-Y. LI: *On generalization of D'Aurizio-Sándor inequalities involving a parameter*, J. Math. Inequal. **12**:3 (2018), 853-860.

[11] B. MALEŠEVIĆ, M. MIĆOVIĆ: *Exponential Polynomials and Stratification in the Theory of Analytic Inequalities*, Journal of Science and Arts. **23**:3 (2023), 659-670.

[12] M. MIĆOVIĆ, B. MALEŠEVIĆ: *Jordan-Type Inequalities and Stratification*, Axioms. **13**:4, 262 (2024), 1-25.

[13] B. MALEŠEVIĆ, D. JOVANOVIĆ: *Frame's Types of Inequalities and Stratification*, Cubo. **26**:1 (2024), 1-19.

[14] B. BANJAC, B. MALEŠEVIĆ, M. MIĆOVIĆ, B. Mihailović, M. SAVATOVIĆ: *The best possible constants approach for Wilker-Cusa-Huygens inequalities via stratification*, Appl. Anal. Discrete Math. **18**:1 (2024), 244-288.

[15] S. CHEN, X. GE: *A solution to an open problem for Wilker-type inequalities*, J. Math. Inequal. **15**:1 (2021), 59-65.

[16] W.-D. JIANG: *New sharp inequalities of Mitrinović-Adamović type*, Appl. Anal. Discrete Math. **17**:1 (2023), 76-91.

[17] R. SHINDE, C. CHESNEAU, N. DARKUNDE, S. GHODECHOR, A. LAGAD: *Revisit of an Improved Wilker Type Inequality*, Pan-American Journal of Mathematics **2** (2023), 1-17.

[18] Y. J. BAGUL, R. M. DHAIGUDE, M. KOSTIĆ, C. CHESNEAU: *Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions*, Axioms. **10**:4, 308, (2021), 1-10.

[19] D.Q. HUY, P.T. HIEU, D.T.T. VAN: *New sharp bounds for sinc and hyperbolic sinc functions via cos and cosh functions*, Afr. Mat. **35**:38 (2024), 1-13.

[20] L. ZHU, R. ZHANG: *New Mitrinović-Adamović type inequalities*, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. **116**:34 (2022), 1-15.

[21] L. ZHU: *New inequalities of Wilker's type for circular functions*, AIMS Mathematics. **5**:5 (2020), 4874-4888.