Two-sided estimates of gamma-function on the real semiaxis

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.
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