Fungrim entry: 306699

\left(x > {x}_{0}\right) \;\implies\; \left(\frac{d^{n}}{{d x}^{n}} \left[\log G(x)\right] > 0\right)\; \text{ where } {x}_{0} = \begin{cases} 3, & n = 0\\2.557664, & n = 1\\1.925864, & n = 2\\0, & n = 3\\ \end{cases}

Assumptions:

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

TeX:

\left(x > {x}_{0}\right) \;\implies\; \left(\frac{d^{n}}{{d x}^{n}} \left[\log G(x)\right] > 0\right)\; \text{ where } {x}_{0} = \begin{cases} 3, & n = 0\\2.557664, & n = 1\\1.925864, & n = 2\\0, & n = 3\\ \end{cases}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; n \in \left\{0, 1, 2, 3\right\}

Definitions:

Fungrim symbol	Notation	Short description
`RealDerivative`	$\frac{d}{d x}\, f\!\left(x\right)$	Real derivative
`LogBarnesG`	$\log G(z)$	Logarithmic Barnes G-function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("306699"),
    Formula(Where(Implies(Greater(x, Subscript(x, 0)), Greater(RealDerivative(Brackets(LogBarnesG(x)), For(x, x, n)), 0)), Equal(Subscript(x, 0), Cases(Tuple(3, Equal(n, 0)), Tuple(Decimal("2.557664"), Equal(n, 1)), Tuple(Decimal("1.925864"), Equal(n, 2)), Tuple(0, Equal(n, 3)))))),
    Variables(x, n),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(n, Set(0, 1, 2, 3)))))

Topics using this entry

Barnes G-function