The Blumenthal Conjecture
نویسندگان
چکیده
The following conjecture was posed by Hayman in his collection of research problems [6]. Let f1(z), f2(z) be entire functions. Is it true that if (1) M(r, f1) = M(r, f2) , 0 < r < ∞, then f1(z), f2(z) are equivalent apart from rotations and reflections? Hayman suggested that the corresponding problem for polynomials (of degree higher than about 6) is also open but such results have yet to be published. We will prove this result for real entire functions with f(0)f ′(0) = 0 and for polynomials with at the most four terms. This set of polynomials includes all quadratic and cubic polynomials.
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