Characteristics of power growth of a multidimensional series of exponents (Q1905115)

Consider a multiple exponent series \[ G(z) = G(z_1, \dots, z_n) = \sum_{p = 1}^\infty D_p \exp \bigl( \langle z, \lambda_p \rangle \bigr), \tag{1} \] where \(D_p \in \mathbb{C}\), \(\{\lambda_p^{(k)}\}\) \((k = 1, \dots, n)\) are sequences of positive numbers verifying \(\lim \{\sum_{k = 1}^n \lambda_p^{(k)}\}^{-1} \ln p = 0\) \((p \to + \infty)\) and \(\langle z, \lambda_p \rangle\) is the scalar product. \textit{V. P. Gromov} [Izv. Akad. Nauk Arm. SSR, Mat. No. 7, 90-103 (1972; Zbl 0238.32001)] found that the domain of convergence of (1) is of the form \(B + i \mathbb{R}^n \subset \mathbb{C}^n\), where \(B\) is a convex octantoid base in \(\mathbb{R}^n\). Let \[ M(G,x) = M(G, x_1, \dots, x_n) = \sup \biggl\{ \bigl |G(x + iy) \bigr |\mid y \in \mathbb{R}^n \biggr\}, \] \[ Q(a) = Q(a_1, \dots, a_n) = \bigl\{ x \in B \mid x_k < a_k,\;a \in \partial B \bigr\},\;|\lambda_p |= \sum^n_1 \lambda_p^{(k)}, \] \[ u_k = a_k - x_k > 0,\;|u |= \sum^n_1 u_k^{-1},\;d = \sqrt {\sum^n_1 u^2_k};\;|\lambda_p |= \sum^n_1 \sqrt {\lambda_p^{(k)}}. \] The author proves that \[ \varlimsup_{d \to 0 +} {\ln M(G,x) \over \ln u} = \varlimsup_{p \to + \infty} \ln^{-1} |\lambda_p |\ln^+ (|D_p |\exp (\langle a, \lambda_p \rangle) \tag{2} \] if \(\lim \ln^{-1} \ln |\lambda_p |\ln \ln p = \beta_0 < 1\) \((p \to + \infty)\). Denote the left-hand side of (2) by \(\rho\). If \(\rho \in (0,+ \infty)\), then \[ \varlimsup_{d\to 0^+} {M(G,x) \over |u |^\rho} = \left( {\rho \over e} \right)^\rho \varlimsup_{p\to + \infty} {D_p \exp \bigl( \langle a, \lambda_p \rangle \bigr) \over |\lambda_p |^{2\rho}}. \]