Motivation: We know that deficient values of an entire function $f$ are important because there is a connection with the singular values of $f$ and hence a connection with the Fatou set of $f$.
There are some sufficient conditions for an entire function has a finite deficient value e.g., if the order $\rho$ of an entire function $f$ is such that $2<\rho<+\infty$ with all but finitely many zeros of $f$ are real, then $0$ is a deficient value of $f$.
Question: Is there any sufficient condition for an entire function of finite order 1 to have a finite deficient value?
In particular, does the function $f(z)=e^z+P(z)$ where $P(z)$ is a complex polynomial have a finite deficient value ?
Target: If I can show that $e^z+P(z)$ has a finite deficient value then all the Fatou components are simply connected.