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Fungrim entry: 99a9c6

logΓ(x)<(x12)log(x)x+log ⁣(2π)2+k=12n+1B2k2k(2k1)x2k1\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(0,)andnZ0x \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
LogGammalogΓ(z)\log \Gamma(z) Logarithmic gamma function
Loglog(z)\log(z) Natural logarithm
ConstPiπ\pi The constant pi (3.14...)
Sumnf(n)\sum_{n} f(n) Sum
BernoulliBBnB_{n} Bernoulli number
Powab{a}^{b} Power
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\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, ConstPi)), 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."))

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

2019-10-05 13:11:19.856591 UTC