Jordan-Type Inequalities and Stratification

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan's inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants.


Introduction
The function: has numerous applications in mathematics.The basic approximation of the sinc x function is given by the well-known Jordan's inequality: , it holds: (1) 2 π ≤ sin x x < 1 .
Since then, many authors have worked on extensions and improvements of Jordan's inequality [1]- [5], [8]- [23].In [8], F. Qi, D.-W.Niu and B.-N. Guo did the elaborate research, summarizing previously discovered improvements and applications of Jordan's inequality, along with related problems.Motivated by some of the following results, this paper provides an additional contribution to this topic.
F. Qi and B.-N. Guo, in the paper [1], provided an enhancement of Jordan's inequality through the following assertion: . Then, it holds: F. Qi then, in the paper [2], provided further improvement of Jordan's inequality through the following assertion: . Then, it holds: In the paper [3], K. Deng contributed to improvements of Jordan's inequality by proving: . Then, it holds: Based on the inequality (3), W. D. Jiang and H. Yun provided further extension of Jordan's inequality in their paper [4] through the following theorem: . Then, it holds: Shortly afterwards, in the paper [5], J.-L.Li and Y.-L.Li provided a more general statement that encompasses the previous inequalities, (2), (3), ( 4) and (5), introducing an entire family of inequalities.Namely, the theorem holds: . Then, it holds: Inspired by Theorems 2, 3, 4, 5 and 6, in this paper, based on the concept of stratification of corresponding families of functions from the paper [6], we introduce a new extension of Jordan's inequality.Namely, by applying stratification, it is possible to extend the inequality (7) so that the parameter n can be a positive real number.The extension of inequalities for real parameters has recently been the subject of various studies [24]- [27], see also [28]- [31].Additionally, we provide the best constants for this type of Jordan's inequality, as well as an analysis of upper and lower bounds and minimax approximations of the sinc x function based on the inequalities (2), (3), ( 4), (5), as well as on the newly obtained inequalities.

Preliminaries
Recently, in the paper [6], the authors considered families of functions φp(x), where x ∈ (a, b) ⊆ R + and p ∈ R + , which are monotonic with respect to the parameter p.In that paper, such families of functions are referred to as stratified families of functions with respect to the parameter p. If, for each x ∈ (a, b), it holds: then the family of functions φp(x) is increasingly stratified with respect to the parameter p. If, for each x ∈ (a, b), it holds: then the family of functions φp(x) is decreasingly stratified with respect to the parameter p.
If it is possible to determine a value of the parameter p = p0 ∈ R + for which the infimum of the error: is attained, then the function φp 0 (x) is the minimax approximant of the family of functions φp(x) on the interval (a, b).Based on the stratifiedness, the parameter value p = p0 is unique.
L'Hôpital's rule for monotonicity was described by the author I. Pinelis in the paper [33], see also [34].In this paper, we use the following formulation: Lemma 1 (Monotone form of L'Hôpital's rule).Let f and g be continuous functions differentiable on (a, b).Suppose f (a+) = g(a+) = 0 or f (b−) = g(b−) = 0, and assume that g ′ (x) ̸ = 0 for all x ∈ (a, b).If f ′ /g ′ is an increasing (decreasing) function on (a, b), then so is f /g.
The method to prove inequalities of the form f (x) > 0 on the interval (a, b) ⊆ R, where f (x) is an MTP function, as outlined in [32], is based on determining a downward polynomial approximation P (x) with respect to the observed function f (x).In [32], the determination of a polynomial P (x) as a polynomial with rational coefficients is considered.If there exists a polynomial P (x) such that f (x) > P (x) and P (x) > 0 on the interval (a, b), then f (x) > 0 holds on the interval (a, b).The polynomial P (x) > 0 is determined as a polynomial with rational coefficients and is examined on the interval (a, b) with rational endpoints.Then, the proof of the inequality P (x) > 0 is an algorithmically decidable problem based on Sturm's theorem, see Theorem 4.2 in [35].In this paper, the application of Sturm's theorem will not be necessary for proving polynomial inequalities.

Main results
In this section, several statements are presented and proven, with a special emphasis on the connection between Jordan's inequality and stratification.Particularly, for each family of functions induced by the aforementioned inequality (7), the best approximations derived from the minimax approximants are identified in Statements 1 and 2.

Lemma 2
The two-parameter family of functions: is individually decreasingly stratified both with respect to the parameter p ∈ R + and with respect to the parameter q ∈ R + on the interval (0, π/2).
(ii) The family of functions: is increasingly stratified with respect to the parameter q ∈ R + on the interval (0, π/2).

Proof. (i) Since
, we obtain the one-parameter family of functions: The first derivative of φ A(q),q (x) with respect to q is: It is evident that: ∂φ A(q),q (x) ∂q < 0 on the interval (0, π/2) for q ∈ R + , which concludes the proof.
Hence, the minimax approximant of the family of functions φ A(q),q (x) is: which determines the corresponding (minimax) approximation:
(ii) It is easily seen that lim φ A(q),q (x) = 0 and lim x→π/2− φ A(q),q (x) = 0.In the part (iv) of this proof, it will be shown that each function φ A(q),q (x), for q ∈ (q1, q2), has exactly one maximum and exactly one minimum on the interval (0, π/2) respectively.Hence, the stated inequalities follow.
(iii) The assertion is equivalent to φ A(q),q (x) < 0 for q ≥ 2 and x ∈ (0, π/2).Continuing from the part (i) of this proof, by multiple applications of L'Hôpital's rule, it can be shown that: lim Considering that g(x) is a decreasing function on the interval (0, π/2), we conclude that the function φ A(q),q (x), for q = q2 = 2, does not have a root on the observed Additionally, based on the stratification (Lemma 3), it holds: for q > 2 on the interval (0, π/2).
By analyzing the monotonicity of the functions ∂φ A(q),q (x) ∂x and φ A(q),q (x) for q = q1 and for q = q2, in a similar manner, it can be concluded that the function φ A(q),q (x), for q = q1, has exactly one maximum on (0, π/2), while the function φ A(q),q (x), for q = q2, has exactly one minimum on (0, π/2).
The equation ( 22) can be numerically solved using the Computer Algebra System Maple, yielding in the value of the parameter q = q0 being numerically determined as: which determines the minimax approximant φ A(q 0 ),q 0 (x) of the family of functions φ A(q),q (x).□ Figure 1 illustrates the stratified family of functions φ A(q),q , see (8).Cases for all values of the parameter q ∈ R + are shown, with a special emphasis on the cases with constants obtained in Statement 1. (i) If q ∈ (0, q1], then the upper bounds of the function sin x x are given by: and the constant q1 is the best possible. (ii) If q ∈ (q1, q2), then the equality: has a unique solution x (q) 0 and it holds: (iii) If q ∈ [q2, +∞), then the lower bounds of the function sin x x are given by: and the constant q2 is the best possible.
Hence, the minimax approximant of the family of functions φ B(q),q (x) is: which determines the corresponding (minimax) approximation:
(ii) Continuing from the previous part of the proof, (i), by multiple applications of L'Hôpital's rule, it can be shown that: The function g(x) from ( 24) determines the values of the parameter q for which the family of functions φ B(q),q (x) have extremes or inflection points on the interval (0, π/2).
(iv) It has been established in the part (ii) of the proof for q ∈ (q1, q2).Similarly, the proof holds for q = q2.
The equation ( 27) can be numerically solved using the Computer Algebra System Maple, yielding in the value of the parameter q = q0 being numerically determined as: which determines the minimax approximant φ B(q 0 ),q 0 (x) of the family of functions φ B(q),q (x).□ Figure 2 illustrates the stratified family of functions φ B(q),q , see (9).Cases for all values of the parameter q ∈ R + are shown, with a special emphasis on the cases with constants obtained in Statement 2.

Applications
In this section, we present two applications.The first application is about the improvements and expansions of Theorems 2, 3, 4 and 5.The second application refers to obtaining some approximations of the sinc function based on some upper and lower bounds of this function and minimax approximants of the corresponding families of functions.

Improvements of Theorems 2, 3, 4 and 5
In order to obtain a generalization of all inequalities from Theorems 3, 4, 5 and 6, for the stratified family of functions φp,q(x) from Lemma 2, we considered the values of the parameter p = A(q) = π − 2 π q+1 and p = B(q) = 2 qπ q+1 as functions depending on the parameter q.It is possible to consider the family of functions φp,q(x) from Lemma 2 by fixing either parameter p or q to some real value.For the cases q = 1, q = 2, q = 3 and q = 4, by applying Statement 1 and 2, improvements and extensions of Theorems 2, 3, 4, 5 respectively can be obtained, as will be shown in the following.Particularly, for each family of functions induced by the considered inequalities, the best approximations derived from the minimax approximants are identified in Statements 4, 5, 6 and 7.
In order to improve and extend Theorem 2, we consider the family of functions φp,q(x) for the case q = 1.The family of functions φp,1(x) reduces to: and is decreasingly stratified with respect to the parameter p ∈ R + on the interval (0, π/2), as proven in Lemma 2. For this family, the following statement holds:

For value:
Hence, the minimax approximant of the family of functions φp,1(x) is: which determines the corresponding (minimax) approximation: Proof.(i) The claim follows directly from Statement 1 and based on the stratification.
(iii) The claim follows directly from Statement 2 and based on the stratification.
(iv) It has been proven within the proof (ii).
(v) Note that the infimum of the error d(p) = sup x∈(0,π/2) |φp,1(x)|, for p ∈ (p1, p2), exists and is attained when: The equation ( 35) can be numerically solved using the Computer Algebra System Maple, yielding in the value of the parameter p = p0 being numerically determined as: which determines the minimax approximant φp 0 ,1(x) of the family of functions φp,1(x).□ In order to improve and extend Theorem 3, we consider the family of functions φp,q(x) for the case q = 2.The family of functions φp,2(x) reduces to: and is decreasingly stratified with respect to the parameter p ∈ R + on the interval (0, π/2), as proven in Lemma 2. For this family, the following statement holds: Then, it holds: (ii) If p ∈ (p1, p2), then the equality: and it holds: (iii) If p ∈ [p2, +∞), then: (iv) Each function from the family φp,2(x), for p ∈ (p1, p2], has exactly one minimum at a point m (p) ∈ (0, π/2) on the interval (0, π/2).
(v) The equality: has the solution p = p0, for the parameter p ∈ (p1, p2), numerically determined as: Hence, the minimax approximant of the family of functions φp,2(x) is: which determines the corresponding (minimax) approximation: Proof.(i) The claim follows directly from Statement 2 and based on the stratification.
(ii) Let us examine the monotonicity of functions φp,2(x) for p ∈ (p1, p2) on the interval (0, π/2) in a similar manner as in the proof of Statement 1.The third derivative of φp,2(x) with respect to x is: x 4 , where the function f (x) is an MTP function given by: f (x) = −x 3 cos x + 6x cos x + 3x 2 sin x − 6 sin x .
(v) The equality: has the solution p = p0, for the parameter p ∈ (p1, p2), numerically determined as: p0 = 0.010441 . . . .Hence, the minimax approximant of the family of functions φp,3(x) is: which determines the corresponding (minimax) approximation: Proof.Analogously to the proof of Statement 5. □ In order to improve and extend Theorem 5, we consider the family of functions φp,q(x) for the case q = 4.The family of functions φp,4(x) reduces to: and is decreasingly stratified with respect to the parameter p ∈ R + on the interval (0, π/2), as proven in Lemma 2. For this family, the following statement holds: (i) If p ∈ (0, p1], then: (ii) If p ∈ (p1, p2), then the equality: and it holds: (iv) Each function from the family φp,4(x), for p ∈ (p1, p2], has exactly one minimum at a point m (p) ∈ (0, π/2) on the interval (0, π/2).Hence, the minimax approximant of the family of functions φp,4(x) is: which determines the corresponding (minimax) approximation: Proof.Analogously to the proof of Statement 5. □

Approximations of the sinc function
In this subsection, we provide some approximations of the sinc function and analyze the maximum approximation errors.The previously obtained upper and lower bounds of the sinc function can be used to derive some approximations of this function.Further, more optimal approximations can be obtained through the corresponding minimax approximants.

Minimax approximation
Maximum deviation of the sinc x function from the sinc x function on the interval (0, π/2) on the interval (0, π/2)

Conclusion
In this paper, two double Jordan-type inequalities have been obtained, encompassing the inequalities established in the papers [1]- [5].These inequalities were explored in the context of stratified families of functions, a concept introduced in recent research [6].The introduction of stratified families of functions enables the derivation of known results for specific parameter choices, including the analysis of parameter values previously unknown in the Theory of Analytic Inequalities.Furthermore, we identify parameter values within each examined family of functions for which the function, as a member of that family, exhibits some optimal properties (minimax approximant).Based on these minimax approximants and functions representing the upper and lower bounds of the sinc function, we provided some approximations of the sinc function.Additionally, we analyzed the errors associated with all mentioned approximations.
It is crucial to emphasize that the minimax approximant of the stratified family of functions is the function for which the minimal error in approximations is obtained within the given family of functions.Therefore, identifying these parameter values is significant in the Approximation Theory.
By considering the stratified family of functions individually with respect to two parameters, we were able to analyze Jordan-type inequalities in a unified manner, resulting in both previously established and novel findings.Future research endeavors will focus on extending this approach even further.

Table 2 :
Lower bounds of the sinc x function on the interval (0, π/2)

Table 1 ,
we present some upper bounds of the sinc function derived from Theorems 2, 3, 4 and 5, that is, Statements 4, 5, 6 and 7 and Statements 1 and 2. It is noteworthy that the upper bound from Theorem 3 (the best upper bound from Statement 5) is identical to the best upper bound from Statement 1.

Table 1 :
Upper bounds of the sinc x function on the interval (0, π/2)In Table2, we present some lower bounds of the sinc function derived from Theorems 2, 3, 4 and 5, that is, Statements 4, 5, 6 and 7 and Statements 1 and 2. It is noteworthy that the best lower bound from Statement 1 is identical to the best lower bound from Statement 2.