- Research
- Open access
- Published:
Windschitl type approximation formulas for the gamma function
Journal of Inequalities and Applications volume 2018, Article number: 272 (2018)
Abstract
In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.
1 Introduction
For \(x>0\), the classical Euler’s gamma function Γ and psi (digamma) function ψ are defined by
respectively. The derivatives \(\psi^{\prime }\), \(\psi^{\prime \prime }\), \(\psi^{\prime \prime \prime },\dots \) are known as polygamma functions. The gamma function has various important applications in many branches of science. For this reason, scholars strive to find various better approximations for the factorial or gamma function by using different ideas and techniques, for instance, Ramanujan [1, p. 339], Burnside [2], Gosper [3], Alzer [4], Shi et al. [5], Batir [6, 7], Mortici [8–12], Nemes [13, Corollary 4.1], [14], Qi et al. [15, 16], Feng and Wang [17], Chen [18–21], Yang et al. [22–25], Lu et al. [26–28], Xu et al. [29]. Some properties of the remainders of certain approximations for the gamma function can be found in [4, 16, 23, 30–35].
In this paper, we are interested in Windschitl’s approximation formula (see [36]) given by
As shown in [21, Eq. (3.18)], the rate of Windschitl’s approximation \(W_{0} ( x ) \) converging to \(\Gamma ( x+1 ) \) is like \(x^{-5}\) as \(x\rightarrow \infty \), and like \(x^{-7}\) if one replaces \(W_{0} ( x ) \) with
by an easy check. These show that \(W_{0} ( x ) \) and \(W_{1} ( x ) \) are more accurate approximations for the gamma function. In 2009, Alzer [37] proved that for all \(x>0\),
with the best possible constants \(\alpha =0\) and \(\beta =1/1620\). Recently, Lu, Song and Ma [27] extended Windschitl’s formula to an asymptotic expansion:
as \(n\rightarrow \infty \) with \(a_{7}=1/810,a_{9}=-67/42525,a_{11}=19/8505, \dots \), and proved that there exists an m such that, for every \(x>m\), the double inequality
holds. An explicit formula for determining the coefficients of \(n^{-k}\) (\(n\in \mathbb{N}\)) was given in [19, Theorem 1] by Chen. Another asymptotic expansion
was presented in the same paper [19, Theorem 2].
Let us consider the four new Windschitl type approximation formulas, as \(x\rightarrow \infty \), which are
The aim of this paper is, by investigating the monotonicity and convexity of the functions
to establish some new sharp inequalities between the gamma function \(\Gamma ( x+1 ) \) and Windschitl’s approximation formula \(W_{0} ( x ) \). As a by-product, a concise proof of Alzer inequalities (1.4) is presented, and a strengthening for Lu et al.’s inequalities (1.6) is given.
The rest of this paper is organized as follows. In Sect. 2, three lemmas are given, which are crucial to the proofs of our results. In Sect. 3, five monotonicity and convexity results for the functions constructed from the gamma function and Windschilt’s formula are proved. Some new inequalities between the gamma or factorial functions with Windschilt’s formula are established in Sect. 4. In Sect. 5, numeric comparisons of several better approximation formulas are presented.
2 Lemmas
To prove our results, we need three lemmas as follows.
Lemma 1
The inequalities
hold for \(x>0\).
Proof
The inequality (2.1) was proved in [38, Remark 2.2].
Let
Then we have
Hence, we conclude that
which proves the first inequality of (2.2).
Analogously, we have
It then follows that
which proves the second formula of (2.2). This completes the proof. □
Lemma 2
The inequalities
hold for all \(t>0\).
Proof
The inequalities in question are equivalent to
and
for \(t>0\), respectively.
Expanding into a power series yields
where
We assert that \(p_{5} ( n ) >0\) for \(n\geq 3\), since \(p_{5} ( n ) \) can be written as
which is evidently positive for \(n\geq 3\). Hence \(h_{1} ( t ) >0\) for all \(t>0\). While
where
It is easy to check that
for \(m=n-5\geq 0\), which proves \(h_{2} ( t ) >0\) for \(t>0\). The proof is complete. □
The following lemma offers a simple criterion to determine the sign of a class of special polynomials on given interval contained in \(( 0,\infty ) \) without using Descartes’ Rule of Signs, which plays an important role in studying certain special functions, see, for example, [39, 40]. A series version can be found in [41, 42].
Lemma 3
([39, Lemma 7])
Let \(n\in \mathbb{N}\) and \(m\in \mathbb{N}\cup \{0\}\) with \(n>m\) and let \(P_{n} ( t ) \) be an nth degree polynomial defined by
where \(a_{n},a_{m}>0\), \(a_{i}\geq 0\) for \(0\leq i\leq n-1\) with \(i\neq m\). Then there is a unique number \(t_{m+1}\in ( 0,\infty ) \) satisfying \(P_{n} (t_{m+1}) =0\) such that \(P_{n} ( t ) <0\) for \(t\in ( 0,t_{m+1} ) \) and \(P_{n} ( t ) >0\) for \(t\in ( t_{m+1},\infty ) \).
Consequently, for a given \(t_{0}>0\), if \(P_{n} ( t_{0} ) >0\) then \(P_{n} ( t ) >0\) for \(t\in ( t_{0},\infty ) \) and if \(P_{n} ( t_{0} ) <0\) then \(P_{n} ( t ) <0\) for \(t\in ( 0,t_{0} ) \).
3 Monotonicity and convexity
Theorem 1
T he function
is strictly decreasing and convex on \(( 0,\infty ) \).
Proof
Differentiation yields
Replacing x by \(( x+1/2 ) \) in inequality (2.1) leads to
and using which to \(f_{0}^{\prime \prime } ( x ) \) gives
Simplifying yields
where
Expanding into a power series gives
where the inequality holds due to
for \(n\geq 4\).
It then follows that \(f_{01} ( t ) >0\) for \(t>0\), so \(f_{0}^{\prime \prime } ( x ) >0\) for \(x>0\). This yields \(f_{0}^{\prime } ( x ) <\lim_{x\rightarrow \infty }f_{0}^{ \prime } ( x ) =0\), which proves the desired result. □
Theorem 2
The function
is strictly increasing and concave on \(( 0,\infty ) \).
Proof
Differentiation yields
Since \(\lim_{x\rightarrow \infty }f_{1}^{\ast \prime } ( x ) =0\), it suffices to prove \(f_{1}^{\ast \prime \prime } ( x ) <0\) for \(x>0 \). Replacing x by \(( x+1/2 ) \) in the right-hand side inequality of (2.2) leads to
which indicates that
where
Using the inequality
yields
for \(t>0\), where the inequality holds due to
for \(t>0\). This implies that \(f_{2}^{\prime \prime } ( x ) <0\) for all \(x>0\), and the proof is complete. □
Theorem 3
The function
is strictly increasing and concave on \(( 0,\infty ) \).
Proof
We clearly see that
where \(D ( y ) =\ln ( 1+y ) -y\). By Theorem 2, \(f_{1}^{\ast }\) is strictly increasing and concave on \(( 0,\infty ) \), so if we prove \(x\mapsto D ( y ) \) is strictly increasing and concave on \(( 0,\infty ) \), then so will be \(f_{1}\), and the proof will be complete. Now we easily check that for \(x>0\),
which completes the proof. □
Theorem 4
The function
is strictly decreasing and convex on \(( 0,\infty ) \).
Proof
Differentiation yields
Since \(\lim_{x\rightarrow \infty }f_{2}^{\prime } ( x ) =0\), it suffices to prove \(f_{2}^{\prime \prime } ( x ) >0\) for \(x>0\). Replacing x by \(( x+1/2 ) \) in the left-hand side inequality of (2.2) leads to
and applying which to \(f_{2}^{\prime \prime } ( x ) \) gives
Making a change of variable \(t=1/x\) yields
We distinguish two cases to prove \(f_{21} ( t ) >0\) for all \(t>0\).
Case 1: \(t\geq 1\). Application of inequality (2.3) gives
where
Clearly, \(p_{6} ( t ) >0\) for \(t\geq 1\), so \(f_{21} ( t ) >0\) for \(t\geq 1\).
Case 2: \(0< t<1\). Using inequality (2.4) yields
where
Since the coefficients of polynomial \(q_{6} ( t ) \) satisfy the conditions of Lemma 3 and \(q_{6} ( 1 ) =53{,}681/990>0\), we find that \(q_{6} ( t ) >0\) for \(t\in ( 0,1 ) \), and then \(f_{21} ( t ) >0\) for \(t\in ( 0,1 ) \).
This ends the proof. □
Theorem 5
The function
is strictly decreasing and convex on \([4/3,\infty )\).
Proof
We easily see that
where \(D ( y ) =\ln ( 1+y ) -y\). By Theorem 4, \(f_{2}\) is strictly decreasing and convex on \(( 0, \infty ) \), so if we prove \(x\mapsto D ( y ) \) is strictly increasing and concave on \([4/3,\infty )\), then so will be \(f_{2}^{\ast }\), and the proof will be complete. Now we easily check that for \(x\geq 4/3\),
where the last inequality holds due to
which completes the proof. □
4 Inequalities
As is well known, analytic inequalities [43–45] play a very important role in different branches of modern mathematics. Using the theorems presented in the previous section, we can obtain some new inequalities for the gamma function and factorial function related to Windschitl’s formula.
Corollary 1
Let \(W_{0} ( x ) \) be defined by (1.2). Then the inequalities
hold for all \(x>0\). If \(x\geq \sqrt{33/35}\), then we have
Proof
The first and second inequalities in (4.1) follow directly from the monotonicity of \(f_{0}\), \(f_{2}\) and \(f_{1}^{\ast }\) on \(( 0,\infty ) \) given in Theorems 1, 4 and 2, respectively, due to \(f_{0} ( \infty ) =\) \(f_{2} ( \infty ) =f_{1}^{\ast } ( \infty ) =0\). The third one holds due to a simple inequality \(1+y< e^{y}\) for \(y>0\). The proof of inequalities (4.2) is similar, which completes the proof. □
Using the monotonicity of \(f_{0}\), \(f_{1}^{\ast }\) and \(f_{2}\) on \(( 0,\infty ) \) and noting that
we immediately get the following corollary.
Corollary 2
For \(n\in \mathbb{N}\), the inequalities
hold with the best constants \(\alpha_{0}=e/\sqrt{2\pi \sinh 1} \approx 1.000{,}34\), \(\beta_{1}^{\ast }=1620e/ ( 1621\sqrt{2 \pi \sinh 1} ) \approx 0.999{,}72\) and \(\alpha_{2}=e^{28{,}349/28 {,}350}/\sqrt{2\pi \sinh 1}\approx 1.000{,}30\).
The proof of inequalities (1.4) presented by Alzer [37] seems to be somewhat complicated. With the aid of the first and second inequalities in (4.1), we can give a new and simpler proof.
Proof of inequalities (1.4)
The sufficiency for the inequalities (1.4) to hold for \(x>0\) follows by the first and second inequalities in (4.1). The necessary condition for the left-hand side inequality of (1.4) to hold for \(x>0\) follows from the following relation:
While the necessary condition for the right-hand side of (1.4) to hold for \(x>0\) follows from the limit relation
This completes the proof. □
The following corollary offers a strengthening for Lu et al.’s inequalities (1.6).
Corollary 3
The inequalities
hold for \(x>c\), where \(c=0\) for the second, third and fourth inequalities, while \(c=x_{0}\approx 0.43738\) for the first one, here \(x_{0}\) is the unique solution of the equation
on \(( 0,\infty ) \).
Proof
Clearly, the second and third inequalities of (4.3) follow by the first and second inequalities in (4.1). It remains to prove the first and last inequalities of (4.3).
(i) The last one is equivalent to
or equivalently,
for \(t=1/x>0\). Denote by \(l ( t ) =\ln \sinh t\) and \(t_{2}= ( t+t^{7}/810 ) \). Then by Taylor formula we have
where \(t<\xi <t+t^{7}/810\). Since \(l^{\prime \prime \prime } ( t ) =2 ( \cosh t ) /\sinh^{3}t>0\), we get
where
Due to
we conclude that \(h_{3} ( t ) >0\) for \(t>0\).
(ii) To ensure that the first inequality holds, it is necessary to establish
for \(x>0\), for which it suffices so show that
By Lemma 3, the numerator in the above fraction, as an 8th degree polynomial, has a unique zero \(x_{0}\) on \(( 0,\infty ) \). Numeric computation gives \(x_{0}\approx 0.437{,}38\).
Now the first inequality is equivalent to
or equivalently,
for \(t=1/x\in ( 0,1/x_{0} ) \), where \(1/x_{0}\approx 2.28632\) is clearly the unique zero of the polynomial
on \(( 0,\infty ) \). In view of \(l^{\prime \prime } ( t ) =-1/\sinh^{2}t<0\), we have
which completes the proof. □
Remark 1
Clearly, the proof of Corollary 3 can also be regarded as a new proof of Lu et al.’s inequalities (1.6). Moreover, our proof gives the minimum value of m, i.e., \(\min ( m ) =x_{0} \approx 0.437{,}38\), such that the the double inequality (1.6) holds for all \(x>x_{0}\).
5 Numeric comparisons
By the asymptotic expansion listed in [46, Eq. (6.1.40)]
we easily verify that our four approximation formulas \(W_{01} ( x ) \), \(W_{01}^{\ast } ( x ) \), \(W_{02} ( x ) \) and \(W_{02}^{\ast } ( x ) \), defined by (1.8), (1.10), (1.9) and (1.11), respectively, have the following limit relations:
Also, for another approximation formula \(W_{1} ( x ) \) defined by (1.3), we have
Denote the two approximation formulas generated by the double inequality (1.6) by
We have
These, in combination with Corollaries 2, 1 and 3, show that the approximation formula \(W_{02} ( x ) \) given by (1.9) is the best among those listed above, which can be seen from comparison Table 1.
6 Conclusion
In this paper, we provide four Windschitl type approximation formulas for the gamma function, and prove that those functions, involving the gamma function and Windschitl type functions, have good properties, including monotonicity and convexity. From these facts we obtain some new sharp Windschitl type bounds for the gamma and factorial functions. These sharp inequalities, together with numerical comparisons, illustrate that \(W_{02} ( x ) \) defined by (1.9) is the best approximation formula among those mentioned in Sect. 5.
Moreover, we give a simple proof of Alzer’s inequalities (1.4), and improve and strengthen Lu et al.’s inequalities (1.6).
It is worth mentioning that our proofs of Theorems 1–5 are subtle and interesting, since the approximations deal with the gamma and hyperbolic sine functions, and it is difficult to establish their monotonicity and convexity by usual methods. Evidently, Lemmas 2 and 3 play important roles.
References
Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Springer, Berlin (1988)
Burnside, W.: A rapidly convergent series for \(\log N!\). Messenger Math. 46, 157–159 (1917)
Gosper, R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40–42 (1978). https://doi.org/10.1073/pnas.75.1.40
Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66(217), 373–389 (1997). https://doi.org/10.1090/S0025-5718-97-00807-7
Shi, X., Liu, F., Hu, M.: A new asymptotic series for the Gamma function. J. Comput. Appl. Math. 195, 134–154 (2006). https://doi.org/10.1016/j.cam.2005.03.081
Batir, N.: Inequalities for the gamma function. Arch. Math. 91, 554–563 (2008). https://doi.org/10.1007/s00013-008-2856-9
Batir, N.: Bounds for the Gamma function. Results Math. 72, 865–874 (2017). https://doi.org/10.1007/s00025-017-0698-0
Mortici, C.: A new Stirling series as continued fraction. Numer. Algorithms 56(1), 17–26 (2011). https://doi.org/10.1007/s11075-010-9370-4
Mortici, C.: Improved asymptotic formulas for the gamma function. Comput. Math. Appl. 61, 3364–3369 (2011). https://doi.org/10.1016/j.camwa.2011.04.036
Mortici, C.: Further improvements of some double inequalities for bounding the gamma function. Math. Comput. Model. 57, 1360–1363 (2013). https://doi.org/10.1016/j.mcm.2012.11.020
Mortici, C.: A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402, 405–410 (2013). https://doi.org/10.1016/j.jmaa.2012.11.023
Mortici, C.: A new fast asymptotic series for the gamma function. Ramanujan J. 38, 549–559 (2015). https://doi.org/10.1007/s11139-014-9589-0
Nemes, G.: New asymptotic expansion for the Gamma function. Arch. Math. (Basel) 95, 161–169 (2010). https://doi.org/10.1007/s00013-010-0146-9
Nemes, G.: More accurate approximations for the gamma function. Thai J. Math. 9(1), 21–28 (2011)
Guo, B.N., Zhang, Y.J., Qi, F.: Refinements and sharpenings of some double inequalities for bounding the gamma function. J. Inequal. Pure Appl. Math. 9(1), Article ID 17 (2008)
Qi, F.: Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function. J. Comput. Appl. Math. 268, 155–167 (2014). https://doi.org/10.1016/j.cam.2014.03.004
Feng, L., Wang, W.: Two families of approximations for the gamma function. Numer. Algorithms 64, 403–416 (2013). https://doi.org/10.1007/s11075-012-9671-x
Chen, C.-P.: Unified treatment of several asymptotic formulas for the gamma function. Numer. Algorithms 64, 311–319 (2013). https://doi.org/10.1007/s11075-012-9667-6
Chen, C.-P.: Asymptotic expansions of the gamma function related to Windschitl’s formula. Appl. Math. Comput. 245, 174–180 (2014). https://doi.org/10.1016/j.amc.2014.07.080
Chen, C.-P., Paris, R.B.: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function. Appl. Math. Comput. 250, 514–529 (2015). https://doi.org/10.1016/j.amc.2014.11.010
Chen, C.-P.: A more accurate approximation for the gamma function. J. Number Theory 164, 417–428 (2016). https://doi.org/10.1016/j.jnt.2015.11.007
Yang, Z.-H., Chu, Y.-M.: Asymptotic formulas for gamma function with applications. Appl. Math. Comput. 270, 665–680 (2015). https://doi.org/10.1016/j.amc.2015.08.087
Yang, Z.-H.: Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function. J. Math. Anal. Appl. 441, 549–564 (2016). https://doi.org/10.1016/j.jmaa.2016.04.029
Yang, Z.-H., Tian, J.-F.: Monotonicity and inequalities for the gamma function. J. Inequal. Appl. 2017, Article ID 317 (2017). https://doi.org/10.1186/s13660-017-1591-9
Yang, Z.-H., Tian, J.-F.: An accurate approximation formula for gamma function. J. Inequal. Appl. 2018, Article ID 56 (2018). https://doi.org/10.1186/s13660-018-1646-6
Lu, D.: A new sharp approximation for the Gamma function related to Burnside’s formula. Ramanujan J. 35(1), 121–129 (2014). https://doi.org/10.1007/s11139-013-9534-7
Lu, D., Song, L., Ma, C.: A generated approximation of the gamma function related to Windschitl’s formula. J. Number Theory 140, 215–225 (2014). https://doi.org/10.1016/j.jnt.2014.01.023
Lu, D., Song, L., Ma, C.: Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula. Appl. Math. Comput. 253, 1–7 (2015). https://doi.org/10.1016/j.amc.2014.12.077
Xu, A., Hu, Y., Tang, P.: Asymptotic expansions for the gamma function. J. Number Theory 169, 134–143 (2016). https://doi.org/10.1016/j.jnt.2016.05.020
Qi, F., Niu, D.-W., Guo, B.-N.: Monotonic properties of differences for remainders of psi function. Int. J. Pure Appl. Math. Sci. 4(1), 59–66 (2007)
Guo, S., Qi, F.: A class of completely monotonic functions related to the remainder of Binet’s formula with applications. Tamsui Oxf. J. Inf. Math. Sci. 25(1), 9–14 (2009)
Shi, X.-Q., Liu, F.-S., Qu, H.-M.: The Burnside approximation of gamma function. Anal. Appl. 8(3), 315–322 (2010). https://doi.org/10.1142/S0219530510001643
Qi, F., Guo, B.-N.: Some properties of extended remainder of Binet’s first formula for logarithm of gamma function. Math. Slovaca 60(4), 461–470 (2010). https://doi.org/10.2478/s12175-010-0025-7
Qi, F., Guo, B.-N.: Integral representations and complete monotonicity of remainders of the Binet and Stirling formulas for the gamma function. Rev. R. Acad. Cienc. Exactas FÃs. Nat., Ser. A Mat. 111(2), 425–434 (2017). https://doi.org/10.1007/s13398-016-0302-6
Yang, Z.-H., Tian, J.-F.: Two asymptotic expansions for gamma function related to Windschitl’s formula. Open Math. 16, 1048–1060 (2018). https://doi.org/10.1515/math-2018-0088
Smith, W.D.: The gamma function revisited. http://schule.bayernport.com/gamma/gamma05.pdf (2006)
Alzer, H.: Sharp upper and lower bounds for the gamma function. Proc. R. Soc. Edinb. 139A, 709–718 (2009). https://doi.org/10.1017/S0308210508000644
Yang, Z.-H., Chu, Y.-M., Zhang, X.-H.: Sharp bounds for psi function. Appl. Math. Comput. 268, 1055–1063 (2015). https://doi.org/10.1016/j.amc.2015.07.012
Yang, Z.-H., Chu, Y.-M., Tao, X.-J.: A double inequality for the trigamma function and its applications. Abstr. Appl. Anal. 2014, Article ID 702718 (2014). https://doi.org/10.1155/2014/702718
Yang, Z.-H., Tian, J.: Monotonicity and sharp inequalities related to gamma function. J. Math. Inequal. 12(1), 1–22 (2018). https://doi.org/10.7153/jmi-2018-12-01
Yang, Z.-H., Tian, J.: Convexity and monotonicity for the elliptic integrals of the first kind and applications. arXiv:1705.05703 [math.CA]
Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462, 1714–1726 (2018). https://doi.org/10.1016/j.jmaa.2018.03.005
Tian, J., Wang, W., Cheung, W.-S.: Periodic boundary value problems for first-order impulsive difference equations with time delay. Adv. Differ. Equ. 2018, Article ID 79 (2018). https://doi.org/10.1186/s13662-018-1539-5
Tian, J.-F.: Triple Diamond-Alpha integral and Hölder-type inequalities. J. Inequal. Appl. 2018, Article ID 111 (2018). https://doi.org/10.1186/s13660-018-1704-0
Tian, J.-F., Ha, M.-H.: Extensions of Hölder’s inequality via pseudo-integral. Math. Probl. Eng. 2018, Article ID 4080619 (2018). https://doi.org/10.1155/2018/4080619
Abramowttz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1972). https://doi.org/10.1063/1.3047921
Acknowledgements
The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.
Funding
This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29) and the Higher School Science Research Funds of Hebei Province of China (No. Z2015137).
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Yang, ZH., Tian, JF. Windschitl type approximation formulas for the gamma function. J Inequal Appl 2018, 272 (2018). https://doi.org/10.1186/s13660-018-1870-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1870-0