Table 2 The H ⁻¹ ( x ; ξ ) function

$\frac{{\left[log\left(x+1\right)\right]}^{1/\beta }}{\lambda }$

[log(x+1)]^1/β

$\lambda {\left[log\left(\frac{x}{\lambda }+1\right)\right]}^{1/\beta }$

σ log(x)+μ

$\frac{x^{1/\beta }\sigma +\mu }{x^{1/\beta }+1}$

$\beta {\left[{(x+1)}^{1/\theta }-1\right]}^{1/{\alpha }_1}$

$\frac{log\left(\beta x+1\right)}{\beta }$

${\left[\frac{log\left(x+1\right)}{log\left(a\right)}\right]}^{1/\beta }$