Hayman’s theorem for analytic functions in a complete Reinhardt domain

Abstract

For functions analytic in a complete multiple circular domain $\mathbb{G}\subset\mathbb{C}^n$ there are established a counterpart of Haymans' Theorem. It specifies that in the definition of boundedness of $\mathbf{L}$-index in joint variables the factorials in the denominator can be removed: An analytic function~$F$ in~$\mathbb{G}$ has bounded $\mathbf{L}$-index in  joint variables  if and only if there exist $p\in\mathbb{Z}_+$ and $c\in\mathbb{R}_{+}$  such that for each  $z\in\mathbb{G}$   $\displaystyle  \max\left\{\frac{|F^{(J)}(z)|}{\mathbf{L}^J(z)}\colon  \|J\|=p+1 \right\}\leq c\cdot   \max\left\{\frac{|F^{(K)}(z)|}{\mathbf{L}^K(z)}\colon  \|K\|\leq p \right\},$   where for $K=(k_1,\ldots,k_n)\in\mathbb{Z}^n_+\colon$ $\|K\|=k_1+\ldots +k_n$,     $\displaystyle  F^{(K)}(z)=\frac{\partial^{\|K\|} F}{\partial z^{K}}(z)= \frac{\partial^{k_1+k_2+\ldots+k_n}H}{\partial z_1^{k_1}\partial z_2^{k_2}\ldots \partial z_n^{k_n}}(z_1, z_2, \ldots, z_n),$   $\mathbf{L}^{K}(z)=l_1^{k_1}(z)\cdot\ldots \cdot l_n^{k_n}(z),$  and the continuous mapping $\mathbf{L}=(l_1(z),l_2(z),\ldots,l_n(z))\colon \mathbb{G}\to \mathbb{R}^n_+$ is locally regularly varying in some sense. It allows to apply this statement in study of local properties of analytic solutions for system of linear higher order partial differential equations. Other result concern estimate of sum  of first $N$ expressions from the definition by the sum of all next expressions of such form $|F^{(K)}(z)|/(K!\mathbf{L}^{K}(z))$, where  $K!=(k_1,\ldots,k_n)$ for $K=(k_1,\ldots,k_n)\in\mathbb{Z}^n_+,$  and $N$ is the $\mathbf{L}$-index in joint variables of the function $F$