We define an entire function on $\mathbb{C}^m$ by $$ f(z_1,\cdots,z_m)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}t^{2n}(z_1^2+\cdots+z_m^2)^n, $$ here $t$ is some (positive) real number. Of course, $f(x)=\text{cos} (t|x|)$ for $x=(x_1,\cdots,x_m)\in\mathbb{R}^m$ hence is bounded by the constant $1$. My question is:

For $x=(x_1,\cdots,x_m)\in\mathbb{R}^m$ and $z=(z_1,\cdots,z_m)\in \mathbb{C}^m$, if $|z_k-x_k|=1, k=1,\cdots, m$, then could we get a reasonable bound for $|f(z)|$ in terms of $t$ and $x=(x_1,\cdots,x_m)$?

I hope it can be bounded by a polynomial of $(t, x)$.