On domains of convergence of series of several complex variables.

Question

I know that for every series of several complex variables $$\sum_{k_1+...+k_n=0}^\infty a_{k_1,...,k_n}z_1^{k_1}\cdots z_n^{k_n}$$ its domain of convergence $G$ is logarithmically convex (that is, $\log G=\{(\log z_1,...,\log z_n):z_i\in G\}$ is a convex set in $\mathbb{R}^{2n}$) Reinhardt domain. But for a series $$\sum_{\mu,\nu=0}^\infty a_{\mu\nu}z_1^\mu z_2^\nu$$ we can describe it's domain of convergence as $$ \begin {cases} |z_1|=\varphi(|z_1|),&& 0\leq|z_1|\leq R\\ |z_2|=R,&&0\leq|z_2|<\varphi(R) \end{cases},$$ where $\varphi$ is continuous and non-decreasing. I know that this set has to be logarithmically convex Reinhardt domain, so what can be said about $\varphi$ to guarantee that is the case? Is it enough to demand that $$\log\circ\varphi$$ is a convex function?