A class of Newton maps with Julia sets of Lebesgue measure zero

Let $g\left(z\right)={\int }_0^{z}p\left(t\right)\exp \left(q\right(t\left)\right)dt+c$ where $p$ and $q$ are polynomials and $c\in C$. Let $f$ be the function derived from Newton's method for $g$. We demonstrate that, under suitable assumptions on the zeros of $g$, the Julia set of $f$ has Lebesgue measure zero. Building on a theorem by Bergweiler, our results imply that $f^n\left(z\right)$ converges to the zeros of $g$ almost everywhere in $C$ provided this is true for each zero of $g$that is not a zero of $g^{\prime }$. To establish these conclusions, we provide general conditions that guarantee the Julia sets of $f$have Lebesgue measure zero.