Geodesic
Two-sided estimates of gamma-function on the real semiaxis
Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 205-221

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In this paper we present new two-sided estimates of gamma-function $\Gamma(x+1)$ on the real semiaxis $x>0$. Based on this result, we improve well-known estimates for the factorial $n!$, which hold for all $n \geq 1$. Some of obtained estimates of gamma-function $\Gamma(x+1)$ hold only for $x \geq 1/2$ and some only for $x \geq 1$. The main estimates are connected to the notion of alternation round of a function by asymptotic series in the strong sense. However such a strong alternation is proved only for several partial sums. We have a conjecture that the asymptotic series alternates round a logarithm of gamma-function in strong sense. Similary we propose new inequalities for the number of $n$-combination from $2n$. These considerations indicate that next investigation is promissing and give a method for construction of new two-sided estimates for functions having alternating asymptotic series. Bibliography: 15 titles.
Keywords: gamma-function, two-sided estimates, asymptotic behavior. 
@article{CHEB_2017_18_2_a12,
     author = {A. Yu. Popov},
     title = {Two-sided estimates of gamma-function on the real semiaxis},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {205--221},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a12/}
}
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A. Yu. Popov. Two-sided estimates of gamma-function on the real semiaxis. Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 205-221. http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a12/