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Let    $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$  g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where  p ,  q  are polynomials and    $$c\in {\mathbb {C}}$$  c ∈ C , and let  f  be the function from Newton’s method for  g . We show that under suitable assumptions on the zeros of    $$g''$$  g ′ ′ the Julia set of  f  has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that    $$f^n(z)$$  f n ( z ) converges to zeros of  g  almost everywhere in    $${\mathbb {C}}$$  C if this is the case for each zero of    $$g''$$  g ′ ′ that is not a zero of  g  or    $$g'$$  g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

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