Fungrim entry: 4a3612

\log G\!\left(x + 1\right) < \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A)

Assumptions:

x \in \left(0, \infty\right)

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\log G\!\left(x + 1\right) < \left(\frac{{x}^{2}}{2} - \frac{1}{12}\right) \log(x) - \frac{3 {x}^{2}}{4} + \frac{\log\!\left(2 \pi\right)}{2} x + \frac{1}{12} - \log(A)

x \in \left(0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("4a3612"),
    Formula(Less(LogBarnesG(Add(x, 1)), Sub(Add(Add(Sub(Mul(Sub(Div(Pow(x, 2), 2), Div(1, 12)), Log(x)), Div(Mul(3, Pow(x, 2)), 4)), Mul(Div(Log(Mul(2, Pi)), 2), x)), Div(1, 12)), Log(ConstGlaisher)))),
    Variables(x),
    Assumptions(Element(x, OpenInterval(0, Infinity))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

Topics using this entry

Barnes G-function