SHARP BOUNDS FOR GAMMA AND DIGAMMA FUNCTION ARISING FROM BURNSIDE’S FORMULA

. The main aim of this paper is to improve the Burnside’s formula for approximating the factorial function. We prove the complete monotonicity of a function involving the gamma function to establish new lower and upper sharp bounds for the gamma and digamma function.


INTRODUCTION
There are many situations of practical problems from pure mathematics, or other branches of science when we are forced to deal with large factorials.As a direct computation cannot be made even by the computer programs, approximation formulas were constructed, one of the most known and most used being the Stirling's formula: It was first discovered in 1733 by the French mathematician Abraham de Moivre (1667-1754) in the form with missing constant.Afterwards, the Scottish mathematician James Stirling (1692-1770) found the constant √ 2π in formula (1.1).For proofs, interesting historical facts, and other details see [8,10].
If in probability theory, or statistics, such approximations are satisfactory, in pure mathematics, more precise estimates are required.Although in the last decades, many authors are concerned to give new improvements of the Stirling's formula, we mention here the following approximation due to W.
In this paper, we study the complete monotonicity of the function f : [0, ∞) → R, given by the formula As a direct consequence, we establish new lower and upper sharp bounds for the gamma and digamma function.More precisely, we prove that for x ≥ 1, , .. is the Euler-Mascheroni constant.Our new inequality improves other known results [2,4,5,6,9] of the form ln x − 1 x < ψ (x) < ln x − 1 2x , x > 1.Similar techniques were developed by the author in the very recent paper [7].

THE RESULTS
The gamma Γ and digamma ψ functions are defined by for every positive real numbers x.The gamma function is an extension of the factorial function, since Γ (n + 1) = n!, for n = 0, 1, 2, 3 . ... The derivatives ψ , ψ , . .., known as polygamma functions, have the following integral representations: (2.1) for n = 1, 2, 3, . ... For proofs and other details, see for example, [1,11].We also use the following integral representation (2.2) Recall that a function f is (strictly) completely monotonic in an interval I if f has derivatives of all orders in I such that (−1) n f (n) (x) ≥ 0, (respective (−1) n f (n) (x) > 0) for all x ∈ I and n = 0, 1, 2, 3 . ... It to be noticed that every non-constant, completely monotonic function is in fact strictly completely monotonic.
Completely monotonic functions involving ln Γ (x) are important because they produce bounds for the polygamma functions.The famous Hausdorff-Bernstein-Widder theorem states that a function f is completely monotonic on [0, ∞) if and only if it is a Laplace transform, where µ is a non-negative measure on [0, ∞) such that the integral converges for all x > 0, see [11, pp. 161].Now we are in position to prove the following Theorem 1.Let there be given f : [0, ∞) → R, by ), we obtain .
But we lim x→∞ f (x) = 0, so f (x) > 0 and consequently, f is strictly increasing.Using the fact that lim x→∞ f (x) = 0, we deduce that f < 0. Finally, −f is completely monotonic.
As a direct consequence of the fact that f is strictly increasing, we