Decay of reciprocal gamma function and similar functions

Question

It is known that the reciprocal gamma function $1/\Gamma$ is entire of order 1 meaning, in particular, that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ \left| \frac{1}{\Gamma(c+it)} \right| \leq C_\varepsilon e^{|t|^\varepsilon} \quad t \in \mathbb R. $$ Now consider the function $$ \Gamma_f(c_1+i x_1,c_2 + ix_2) = \int\limits_0^\infty \int\limits_0^\infty y_1^{ix_1+c_1-1} y_2^{ix_2 + c_2 -1} e^{-f(y_1,y_2)} \, dy_1 dy_2, $$ where $f(y_1,y_2)$ is a smooth positive and positively homogeneous of order 1, i.e. $f(\lambda y_1, \lambda y_2) = \lambda f(y_1,y_2)$, $\lambda > 0$; $c_1$, $c_2 > 0$. If $f(y_1,y_2)=y_1+y_2$ then $$ \Gamma_f(c_1+ix_1,c_2+ix_2) = \Gamma(c_1+ix_1)\Gamma(c_2+ix_2) $$ so that for any $\varepsilon > 1$ there exists $C_\varepsilon > 0$ such that $$ |\Gamma_f(c_1 + ix_1, c_2 + ix_2)|^{-1} \leq C_\varepsilon e^{|x|^\varepsilon}, \quad x \in \mathbb R^2. $$ My question is whether this inequality also holds in the case of arbitrary smooth positive and positively homogeneous $f$? If the answer is non-trivial, what techniques should I use to obtain some similar estimate? For gamma function this is obtained using integration by parts which doesn't lead to something good in the case of general $f$.