A note on the values of Barnes $G$-function at rational points

Let $\gamma$ be the Euler's constant, and $\psi(z)$ the digamma function. It is known that $\psi(z)+\gamma$ is transcendental whenever $z\in\mathbb{Q}\backslash\mathbb{Z}$. A classical function which comes close at the heels of the digamma function is the logarithmic derivative of the Barnes $G$-function, denoted by $\psi_G(z)$. Analogous to the above result on $\psi(z)$, we show in this paper that $$\psi_G(z)-(1-z)\gamma-\frac{1}{2}\log 2\pi$$ is transcendental whenever $z\in\mathbb{Q}\backslash\mathbb{Z}$. Further, under Schanuel's conjecture, we show that the above result is true without the term $\frac{1}{2}\log 2\pi$, and at most one element from $$\{\psi_G(z), z\in \mathbb{Q}\backslash\mathbb{Z}\}$$ is algebraic.