Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is complex $n$-dimensional, and $$t_1 = t, t_2 = t^{\rho(x)}, y = \log t.$$ $\rho (x) $ is some given function, $\log$ is the natural logarithm, and $\tilde{\mathbb{C} \setminus \{ 0 \}}$ is the universal covering space of $\mathbb{C} \setminus \{0\}$. Assume that the coefficients are holomorphic on the same disk.
It is known that $u$ converges on the region $$ \{(t,x) : |t_1| < \epsilon, |t_2 | <\epsilon , |yt_1|<\epsilon, |y^2 t_2| <\epsilon, |x_i| < r\ (i =1,...,n) \}.$$
I want to show that $$ \max_{x \in K} |u(t,x)| = O(t^a)$$ as $t$ tends to zero in $S_\theta := \{t : |\arg t|<\theta \}.$
What I have so far: Take $r_1 > 0, r_2 > 0, r_3 > 0,$ and $r_4 > 0$ so that $r_1 < \epsilon, r_2 < \epsilon, r_3 r_1 < \epsilon, r_3^2 r_2 <\epsilon$ and $r_4 < r$ hold. Then $u$ is now convergent on the region $$ \{(t_1, t_2 , x, y) : |t_1| \leq r_1 , |t_2| \leq r_2, |y| \leq r_3, |x_i| \leq r_4\ (i=1,...,n) \}.$$ We can get Cauchy's estimates for the coefficients as follows: For some $M$, $$ |u_i (x) | \leq \dfrac{M}{r_1^i} \qquad \text{and} \qquad |\varphi_{i,j,k} (x) | \leq \dfrac{M}{r_1^i r_2^j r_3^k}$$ on $D(r_4) := \{x : |x_i| \leq r_4\ (i=1,...,n) \}.$
There, any help - hints, suggestions on how to proceed, results that can be used - would be appreciated. Thank you!