Estimating the digamma and trigamma functions by completely monotonicity arguments. (English) Zbl 1207.33002
A real valued function \(f\) is completely monotonic on \((0,1)\) if it has derivatives of all orders and for every \(x\in (0,1)\) and \(n\geq 1\), we have \((-1)^nf^{(n)}(x)\geq 0\). Motivated by approximation of the Gamma function and \(n!\), the author considers the function \(F_a:[0,\infty)\to\mathbb{R}\) defined by
\[ F_a(x)=\ln\Gamma(x+1)-\left(x+\frac{1}{2}\right)\ln(x+a)+x+a+\frac{1}{2}-\ln\sqrt{2\pi e}, \]
and shows that the functions \(F_a\) and \(-F_b\) are completely monotonic, respectively for \(a\in [0,(3-\sqrt{3})/6]\) and \(b\in [1/2,(3+\sqrt{3})/6]\). The author applies this result to obtain some bounds for digamma and trigamma functions.
\[ F_a(x)=\ln\Gamma(x+1)-\left(x+\frac{1}{2}\right)\ln(x+a)+x+a+\frac{1}{2}-\ln\sqrt{2\pi e}, \]
and shows that the functions \(F_a\) and \(-F_b\) are completely monotonic, respectively for \(a\in [0,(3-\sqrt{3})/6]\) and \(b\in [1/2,(3+\sqrt{3})/6]\). The author applies this result to obtain some bounds for digamma and trigamma functions.
Reviewer: Mehdi Hassani (Zanjan)
MSC:
| 33B15 | Gamma, beta and polygamma functions |
| 26D07 | Inequalities involving other types of functions |
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