About some exponential inequalities related to the sinc function

In this paper, we prove some exponential inequalities involving the sinc function. We analyze and prove inequalities with constant exponents and inequalities with certain polynomial exponents. Also, we establish intervals in which these inequalities hold.

The following inequalities are proved in [8]: for every x ∈ (0, π). In [9], the authors considered possible refinements of inequality (1) by a real analytic function ϕ a (x) = ( sin x x ) a for x ∈ (0, π) and parameter a ∈ R and proved the following inequalities: Statement 1 ([9], Theorem 10) For all x ∈ (0, π) and a ∈ (1, 3 2 ), In [9], based on the analysis of the sign of the analytic function F a (x) = sin x x a cos 2 x 2 in the right neighborhood of zero, the corresponding inequalities for parameter values a ≥ 3 2 are discussed. In this paper, in Sect. 3.1, using the power series expansions and the Wu-Debnath theorem, we prove that inequality (2) holds for a = 3 2 . At the same time, this proof represents another proof of Statement 1. Also, we analyze the cases a ∈ ( 3 2 , 2) and a ≥ 2 and prove the corresponding inequalities.
In Sect. 3.2, we introduce and prove a new double-sided inequality of similar type involving polynomial exponents.
Finally, in Sect. 3.3, we establish a relation between the cases of constant and polynomial exponents.

Preliminaries
In this section, we review some results that we use in our study.
In accordance with [10], the following expansions hold: where B i (i ∈ N) are Bernoulli's numbers.
In our proofs, we use the following theorem proved by Wu and Debnath [11]. ). Then, for all x ∈ (a, b), we have the following inequality: Furthermore, if (-1) n f (n) (x) is decreasing on (a, b), then the reversed inequality of (5) holds. increasing on (a, b). Then, for all x ∈ (a, b), we have the following inequality: Furthermore, if f (n) (x) is decreasing on (a, b), then the reversed inequality of (6) holds.
Remark 1 Note that inequalities (5) and (6) hold for n ∈ N and for n = 0.
Here, and throughout this paper, a sum where the upper bound of summation is lower than its lower bound is understood to be zero.
The following theorem, which is a consequence of Theorem WD, was proved in [12]. Theorem 2 ([12], Theorem 1) Let a function f : (a, b) − → R have the following power series expansion: for x ∈ (a, b), where the sequence of coefficients {c k } k∈N 0 has a finite number of nonpositive terms, and their indices are in the set J = {j 0 , . . . , j }. Then, for the function (8) and the sequence {C k } k∈N 0 of the nonnegative coefficients defined by we have for every x ∈ (a, b). Also, F (k) (a+) = k!C k , and the following inequalities hold: for all x ∈ (a, b) and n ∈ N 0 , that is, for all x ∈ (a, b) and m > max{j 0 , . . . , j }.

Inequalities with constants in the exponents
First, we consider a connection between the number of zeros of a real analytic function and some properties of its derivatives. It is well known that the zeros of a nonconstant analytic function are isolated [13]; see also [14] and [15]. We prove the following statement.
Suppose that the following conditions hold: (1) and (2) we conclude that there exists exactly one zero By repeating the described procedure, we get the statement of the theorem.

Theorem 4
For all x ∈ (0, π), we have Since sin x x 3 2 ≤ sin x x a for x ∈ (0, π) and a ∈ (1, 3 2 ], the previous theorem can be thought of as a new proof of Statement 1. Consider now the family of functions f a (x) = a ln sin x x -2 ln cos x 2 for x ∈ (0, π) and parameter a > 3 2 . It easy to check that for the sequence the following equivalences are true: Let us now consider the function m : It is not difficult to check that lim a→2 -m(a) = +∞, whereas for a fixed a ∈ ( 3 2 , 2), the number of negative elements of the sequence {E k } k∈N is m(a), and their indices are in the set {1, . . . , m(a)}. For this reason, we distinguish two cases a ∈ ( 3 2 , 2) and a ≥ 2. As for the parameter a = 2 and x ∈ (0, π), we have whereas for a > 2 and x ∈ (0, π), we have Hence, we have proved the following theorem. Consider now the case where the parameter a ∈ ( 3 2 , 2). As noted before, for any fixed a ∈ ( 3 2 , 2), there is a finite number of negative coefficients in the power series expansion (17), so it is possible to apply Theorem 2.

Lemma 1 Consider the family of functions f a (x) = a ln sin x
x -2 ln cos x 2 for x ∈ (0, π) and parameter a ∈ Proof Let us recall that, for any fixed a ∈ ( 3 2 , 2), there is a finite number of negative coefficients in the power series expansion (17). Also, we have For the derivatives of the function f a (x) in the left neighborhood of π , it suffices to observe that From this the conclusions of the lemma can be directly derived.
Thus, for every a ∈ ( 3 2 , 2), the corresponding function f a (x) = a ln sin x x -2 ln cos x 2 has exactly one zero on the interval (0, π). Let us denote it by x a .
The following theorem is a direct consequence of these considerations.
Remark 7 For a fixed a ∈ ( 3 2 , 2), select n > m(a) + 1 and consider inequalities (22). Denote the corresponding polynomials on the left-and right-hand sides of (22) by P L (x) and P R (x), respectively. These polynomials are of negative sign in a right neighborhood of zero (see [15], Theorem 2.5), and they have positive leading coefficients. Then, the root x a of the equation f a (x) = 0 is always localized between the smallest positive roots of the equations P L (x) = 0 and P R (x) = 0.

Inequalities with polynomial exponents
In this subsection, we propose and prove a new double-sided inequality involving the sinc function with polynomial exponents.
To be more specific, we find two polynomials of the second degree that, when placed in the exponent of the sinc function, give an upper and a lower bound for cos 2 x 2 .
Remark 9 Note that this method can be used to prove that inequality (24) of Theorem 8 holds on any interval (0, c) where c ∈ (0, π), but the degrees of the polynomials H 1 and H 2 get larger as c approaches π .

Constant vs. polynomial exponents
Let us observe the inequalities in Theorems 6 and 8 and inequality (24) containing constants and polynomials in the exponents, respectively.
A question of establishing a relation between these functions, with different types of exponents, comes up naturally. The following theorem addresses this question. Hence, applying Theorem 8, the double-sided inequality (28) holds for all a ∈ ( 3 2 , 2) and x ∈ (0, m a ).
In Table 1 we show the values x a and m a for some specified a ∈ ( 3 2 , 2).

Conclusion
In this paper, using the power series expansions and the application of the Wu-Debnath theorem, we proved that inequality (2) holds for a = 3 2 . At the same time, this proof represents a new short proof of Statement 1.
We analyzed the cases a ∈ ( 3 2 , 2) and a ≥ 2, and we proved the corresponding inequalities. We introduced and proved a new double-sided inequality of similar type involving polynomial exponents. Also, we established a relation between the cases of constant and polynomial exponents.