# Fungrim entry: 2398a1

$\log \Gamma(x) > \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}}$
Assumptions:$x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
References:
• H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8.
TeX:
\log \Gamma(x) > \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n} \frac{B_{2 k}}{2 k \left(2 k - 1\right) {x}^{2 k - 1}}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
LogGamma$\log \Gamma(z)$ Logarithmic gamma function
Log$\log(z)$ Natural logarithm
Pi$\pi$ The constant pi (3.14...)
Sum$\sum_{n} f(n)$ Sum
BernoulliB$B_{n}$ Bernoulli number
Pow${a}^{b}$ Power
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("2398a1"),
Formula(Greater(LogGamma(x), Add(Add(Sub(Mul(Sub(x, Div(1, 2)), Log(x)), x), Div(Log(Mul(2, Pi)), 2)), Sum(Div(BernoulliB(Mul(2, k)), Mul(Mul(Mul(2, k), Sub(Mul(2, k), 1)), Pow(x, Sub(Mul(2, k), 1)))), For(k, 1, Mul(2, n)))))),
Variables(x, n),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, ZZGreaterEqual(0)))),
References("H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66(217), pp. 373-389. Theorem 8."))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC