Fungrim entry: 07584a

R_G\!\left(x, y, z\right) \le \min\!\left(\sqrt{\frac{x + y + z}{3}}, \frac{{x}^{2} + {y}^{2} + {z}^{2}}{3 \sqrt{x y z}}\right)

Assumptions:

x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right)

TeX:

R_G\!\left(x, y, z\right) \le \min\!\left(\sqrt{\frac{x + y + z}{3}}, \frac{{x}^{2} + {y}^{2} + {z}^{2}}{3 \sqrt{x y z}}\right)

x \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; y \in \left[0, \infty\right) \;\mathbin{\operatorname{and}}\; z \in \left[0, \infty\right)

Definitions:

Fungrim symbol	Notation	Short description
`CarlsonRG`	$R_G\!\left(x, y, z\right)$	Carlson symmetric elliptic integral of the second kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("07584a"),
    Formula(LessEqual(CarlsonRG(x, y, z), Min(Sqrt(Div(Add(Add(x, y), z), 3)), Div(Add(Add(Pow(x, 2), Pow(y, 2)), Pow(z, 2)), Mul(3, Sqrt(Mul(Mul(x, y), z))))))),
    Variables(x, y, z),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(y, ClosedOpenInterval(0, Infinity)), Element(z, ClosedOpenInterval(0, Infinity)))))

Topics using this entry

Carlson symmetric elliptic integrals