Let $f$ and $g$ be two non-zero meromorphic functions of finite order, in the sense that two numbers $\rho_f$ and $\rho_g$ exist such that $$f(z)=\mathcal{O}(e^{|z|^{\rho_f}})$$ $$g(z)=\mathcal{O}(e^{|z|^{\rho_g}})$$ when $z$ goes to infinity. Suppose that the quotient $$\frac{f}{g}$$ define an entire function (after removing removable singularities). Is $f/g$ of finite order in the above sense ?
Thanks for your help !
(I thought using Hadamard's factorization theorem but ultimately I doubt that such a result is true)