What I need is a monotonically decreasing function that forms an upper bound for the following function: \begin{equation} - G_{0,6}^{4, 0}\biggl({-\atop -\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{1}{3},\frac{2}{3}}\bigg| x \biggr) \qquad ,x\in\mathbb{R}_+ \end{equation} where G is the Meijer-G function.
By estimating the asymptotes (calculated by Mathematica) for $x\rightarrow \infty$, I found the following candidate: \begin{equation} f(x)=\frac{e^{-3x^{\frac{1}{6}}}\sqrt{\pi}}{96x^{\frac{1}{12}}}\bigl(54+18\sqrt{3}+(96+32\sqrt{3})x^{\frac{1}{6}}\bigr) \end{equation}
The question is: Is it indeed true that $\forall x \in \mathbb{R}_+ \qquad - G_{0,6}^{4, 0}\biggl({-\atop -\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{7}{6},\frac{1}{3},\frac{2}{3}}\bigg| x \biggr)\leq f(x)$ ?
And if so, how could I proof it? If not, are there other ways to find an upper bound?
Maybe one can prove this if one would have an full asymptotic series expansion of the Meijer-G function for $x\rightarrow \infty$. But I do not know how to calculate this.