On entire Dirichlet series similar to Hadamard compositions

A function $F(s)=\sum_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with $0\le\lambda_n\uparrow+\infty$ is called the Hadamard composition of genus $m\ge 1$ functions $F_j(s)=\sum_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$ if $a_n=P(a_{n,1},...,a_{n,p})$, where$P(x_1,...,x_p)=\sum\limits_{k_1+\dots+k_p=m}c_{k_1...k_p}x_1^{k_1}\cdot...\cdot x_p^{k_p}$ a homogeneous polynomial degree 1$. Let $M(\sigma,F)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and $\alpha,\,\beta$ be positive continuous increasing to $+\infty$ on $[x_0, +\infty)$. To characterize growth $M(\sigma,F)$, we use generalized order $\varrho_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\alpha(\ln\,M(\sigma,F))}{\beta(\sigma)}$, type$T_{\alpha,\beta}[F]=\varlimsup\limits_{\sigma\to+\infty}\dfrac{\ln\,M(\sigma,F)}{\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}$and membership in convergence class defined by condition$\displaystyle \int_{\sigma_0}^{\infty}\frac{\ln\,M(\sigma,F)}{\sigma\alpha^{-1}(\varrho_{\alpha,\beta}[F]\beta(\sigma))}d\sigma<+\infty.$Assuming $\alpha, \beta$ $\alpha^{-1}(c\beta(\ln\,x))$ are slowly for each $c\in (0,+\infty)$ $\ln\,n=O(\lambda_n)$ as $n\to \infty$, it proved, example, that $F_j$ have same $\varrho_{\alpha,\beta}[F_j]=\varrho\in types $T_{\alpha,\beta}[F_j]=T_j\in [0,+\infty)$, $c_{m0...0}=c\not=0$, $|a_{n,1}|>0$ $|a_{n,j}|= o(|a_{n,1}|)$ $n\to\infty$ $2\le j\le p$, $F$ genus$m\ge then $\varrho_{\alpha,\beta}[F]=\varrho$ $\displaystyle T_{\alpha,\beta}[F]\le \sum_{k_1+\dots+k_p=m}(k_1T_1+...+k_pT_p).$It proved also belongs only ifall belong class.