Fungrim entry: 99a9c6

\log \Gamma(x) < \left(x - \frac{1}{2}\right) \log(x) - x + \frac{\log\!\left(2 \pi\right)}{2} + \sum_{k=1}^{2 n + 1} \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 + 1} \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("99a9c6"),
    Formula(Less(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, Add(Mul(2, n), 1)))))),
    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

Bounds and inequalities for the gamma function