Some Rational Approximations and Bounds for Bateman’s G -Function

: Symmetrical patterns exist in the nature of inequalities, which play a basic role in theo-retical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman’s G -function G ( w ) = 1 w + (cid:34) 2 w 2 + n ∑ j = 1 4 α j w 2 − 2 j (cid:35) − 1 + O (cid:16) 1 w 2 n + 2 (cid:17) , where α 1 = 14 , and α j = ( 1 − 2 2 j + 2 ) B 2 j + 2 j + 1 + ∑ j − 1 ν = 1 ( 1 − 2 2 j − 2 ν + 2 ) B 2 j − 2 ν + 2 α ν j − ν + 1 , j > 1. As a consequence, we introduced some new bounds of G ( w ) and a completely monotonic function involving it.

Abstract: Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman's G-function G(w) = 1 w + 2w 2 + n ∑ j=1 4α j w 2−2j −1 (1−2 2j−2ν+2 )B 2j−2ν+2 α ν j−ν+1
There are several remarkable applications of completely monotonic functions in many scientific branches; for more information about this topic, we refer to [12][13][14]. The convergence of the integral ( [15], p. 160): where κ(w) is bounded and non-decreasing for w ≥ 0, gives the necessary and sufficient condition for C(w) to be completely monotonic on w ≥ 0.
In [11], Qiu and Vuorinen deduced the double inequality: and Mortici [10] improve it by where γ is the Euler constant. In [5], Mahmoud and Agarwal improved the lower bound of Inequality (5) for w > 9−12 ln 2 16 ln 2−11 Furthermore, they presented the asymptotic formula: where the B n s are the Bernoulli numbers.
In [7], Mahmoud, Talat, and Moustafa studied the following approximation family: which is asymptotically equivalent to G(w) for w → ∞, and they proved the inequality: where a = 4 e 2 −4 and b = 1 are the best possible. In [3] Hegazi, Mahmoud, Talat, and Moustafa deduced that: and where c = 3 and d = e 4 −16 12 are the best possible.
Recently, Mahmoud and Almuashi [8] studied the generalized Bateman's G-function G µ (w) defined by and presented some of its properties. Furthermore, they presented the following inequality: and ι = 1 are the best possible.
The outline of the paper is as follows. Section 1 provides the definition, some relations, asymptotic expansions, and some inequalities of the Bateman G-function. The Padé approximant is defined in Section 2, and some rational approximations of G(w) are calculated. In Section 3, some new bounds of G(w) are presented based on Padé approximants, and we show that our new inequalities improve some recently published ones. We prove the complete monotonicity property of a function involving G(w) in Section 4.

Some Padé Approximants of Bateman's G-Function
In this section, we present some Padé approximants of the function G(w), which present the best rational approximations with the given order of a function.
Consider the formal power series: then the rational function: is called the Padé approximant of order (r, s) of the function h(w) ( [16], Chapter 1; [17], p. 96) and [18] where and the coefficients b i s are the solution of the system: with c i = 0 for i < 0, and the coefficients a i s are given by Proof. For the function: the Padé approximant of order (1, n) is given by where the coefficients b ν s are the solution of the system Then, To obtain the Padé approximant of order (2,16) of the function f (w), we consider the system: and hence, we obtain b 2 = −

Some Rational Bounds of Bateman's G-Function
In this section, we use Padé approximants to formulate new bounds of the function G(w).

Theorem 2.
The following inequality holds: where the lower bound and the upper bound hold for w > 0 and w > √ √ 13−1 2 0.80709, respectively.

Remark 2.
The upper bound of (23) improves the upper bound of (15) for w ∈ R + − √ √ where r 3 (w) = 128,654,692w 12  have no positive real roots, then r 3 (w) and s 3 (w) are positive ∀ w > 0, which shows that the lower bound of (24) improves the lower bound of (15) for w > 0. Furthermore, consider the difference:  16] f (w) > 0 for w > 26 9 , which shows that the upper bound of (24) improves the upper bound of (15) for w > 26 9 .

A Completely Monotonic Function Involving G(w)
In this section, we present another advantage of Padé approximants in formulating new completely monotonic functions involving the function G(w). Theorem 4. The function: is completely monotonic for w > 0, that is Proof. The function M satisfies Using the relation: a n t n 362l880 n! where a n = (n − 2)(31n 4 − 186n 3 + 500n 2 − 150n + 189)(n − 10)(n − 8)(n − 6)(n − 4) we obtain Taylor's formula with the remainder of the function sin t gives the following double inequality ( [20], p. 284) for n ∈ N: where | sin (n+1) t| ≤ 1. Furthermore, using the expansion: The two inequalities (29) and (32) complete the proof.

Conclusions
The Padé approximant method presents some rational approximations for Bateman's G-function G(w). These approximations provided us with new inequalities of the function G(w) with completely monotonic functions involving it. We presented proofs to clarify the novelty of our results, which could be of interest to a large part of the readers. This method is considered a powerful tool in deducing estimates and inequalities for several other special functions.