REVISIT OF AN IMPROVED WILKER TYPE INEQUALITY

A BSTRACT . In this article, we revisit an improved Wilker type inequality established in 2020. We ﬁll some gaps in the existing proof and propose an alternative proof using the same mathematical ingredients. All the details are given for checking purposes.


INTRODUCTION
Trigonometric type inequalities hold significant importance in mathematics and its applications.These inequalities establish bounds and relationships among trigonometric functions and play a fundamental role in solving problems involving angles and periodic functions.They provide valuable tools for proving other mathematical theorems and inequalities, aiding in the development of mathematical analysis and calculus.Moreover, trigonometric type inequalities find practical applications in physics, engineering, and various scientific disciplines, enabling accurate modeling and prediction of phenomena involving periodic behavior.Emphasizing their significance contribute to a deeper understanding and utilization of trigonometry in diverse fields.Former and recent results on this topic can be found in [1][2][3][4][5][6][7][8][9].
In particular, the authors in [2] demonstrated the following inequalities: It is known as the Wilker inequalities.Elegant proofs are given in [10] and [11].In 2020, Theorem 1.1 in [12] stated the following improvement: Theorem 1.1.[12,Theorem 1.1]The inequality, is valid.
The proof of this sharp inequality given on page no.4879 is mainly based on differentiation techniques, several power series formulae and the deep use of Bernoulli's numbers B n (including a result established in [9]).However, most of the details in the developments are omitted, certainly for the sake of place; the developments are very demanding in their algebraic manipulations of numerous terms.Unfortunately, it seems that a term has been omitted at one step, and in all the other steps, making the proof perfectible from a mathematical viewpoint.The aim of this paper is to revisit this proof.In Section 2, we revise some key results of the proof of Theorem 1.1 in [12].The detailed proofs are given in Section 3.
2. ALTERNATIVE PROOF OF THEOREM 1.1 IN [12] The proof of Theorem 1.1 in [12] is mainly based on three complementary results: one on the derivative of a thoroughly selected function, one on a series expansion of a key function and another on an inequality involving the coefficient of the previous series expansion.

Main results.
The next lemma presents the obtained differentiation of the main function used in the first step in the proof of Theorem 1.1 in [12].Lemma 2.1.Let us consider the following function: Then we have In the proof of Theorem 1.1 in [12], it is found that In other words, the term 12x 3 tan 4 x is missing in the derivative of F (x) in the proof of Theorem 1.1 in [12].And this term is not considered in the rest of the proof; it is not just a local miss (see Subsection 2.2).The rest of the proof of Theorem 1.1 in [12] consists in showing that f † (x) > 0 for all x ∈ (0, π/2).This is however not possible; a counter example is given as follows: The term 12x 3 tan 4 x is thus crucial to obtain the valid proof.
The following lemma present a valid power series expansion for f (x) as described in Equation (2.1), with the same notations as in the proof of Theorem 1.1 in [12], which allows us to conclude properly the proof of Theorem 1.1 in [12].
Lemma 2.2.The following power series expansion holds: The difference obtained with the series expansion of f † (x) as obtained in the proof of Theorem 1.1 in [12] is substantial (see Subsection 2.2).To end the proof of Theorem 1.1 in [12], an inequality involving l n must be established, and it is in the next result.
If we combine Lemmas 2.1, 2.2 and 2.3, as in the proof of Theorem 1.1 in [12], by noticing that • w(n) > 0 for any n ≥ 6; a numerical support of this claim is given in Table 1 for the integers 6 to 100, with results divided by 100000 and rounded to the third decimal for the sake of place (in fact, we have w(6) = 271.169455316,w(7) = 1299.11425957,w(8) = 3764.36501028,etc.) • 2 2n > v1(n) u1(n) for all n ≥ 6 by using mathematical induction; a numerical support of this claim is given in Table 2, where the function g(n)/1000000 with is considered for n = 6, 7, . . ., 20.The following lemma shows that the missing term 12x 3 tan 4 x is well not considered in the rest of the proof of Theorem 1.1 in [12]; it is not just a local miss.
Lemma 2.4.For any x ∈ (0, π/2), we can decompose f † (x) as defined in (2.2) as where This result follows the one in the proof of Theorem 1.1 in [12] except the denominator of the first term of l † n (when we solve step by step, we get (2n − 2)! instead of (2n)! as we actually see in the published proof) and the fact that the term 12x 3 tan 4 x is omitted, which is a crucial gap.
The conclusion that F (x) > 0 in the proof of Theorem 1.1 in [12] comes from the fact that the analogous term of w(n) defined by is claimed to be strictly positive when n ≥ 6, whereas it is not, as proved by the following counter example: The rest of the article is devoted to the proof of all the presented lemmas.

PROOFS
For the sake of correctness, check and reproducibility, the proof contains the maximum details.
3.1.Proof of Lemma 2.1.The proof is based on differentiation and technical arrangements.For the sake of correctness, all the details are provided below.We have x 22 (tan x) 6 .
Therefore, we establish that We can develop this expression as follows: Hence, we have where This ends the proof of Lemma 2.1.

Proof of Lemma 2.2.
The proof is based on the same series expansions used in the proof of Theorem 1.1 in [12].We have Therefore, we can write After some arrangements, we get We can write Therefore, we have So, we can write Hence, we find that where This ends the proof of Lemma 2.2.

Proof of Lemma 2.3.
The proof is based on the same inequalities and manipulations of Bernoulli's numbers used in the proof of Theorem 1.1 in [12].We have We know that Hence, we have Therefore, we establish that This implies that As a result, we get .

Thus, we can write
where This ends the proof of Lemma 2.3.

Proof of Lemma 2.4.
The proof used the same techniques as in the one of Lemma 2.2.We have As a result, we have Therefore, we establish that We obtain We can arrange this function as follows:  As a result, we obtain This ends the proof of Lemma 2.4.