Infinite iterated power with alternating coefficients

Let
$$ f(z)=z^{\beta\cdot z^{z^{\beta\cdot z^{z^{\beta\cdot z^{\dotsb}}}}}} $$
where $\beta\in\mathbb C$ and $|\beta|>1$, be an infinite iterated power. Then $f(z)$ is a holomorphic function in some domain $U\supset e^K\cap\{z:|{\arg z}|<\pi\}$, where $e^K$ is the image of the disc $K=\{w:|w|<R\}$ of radius defined by the formula $1/R=\sqrt{|\beta|}\cdot\exp((1+t^2)/(1-t^2))$ and $t=t(\sqrt{|\beta|}\,)\in[0,1)$ is the solution of the equation $\sqrt{|\beta|}=\dfrac{1+t}{1-t}\cdot\exp(2t/(1-t^2))$.