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This paper is concerned with a special class of holomorphic functions with extremal properties of its absolute values on arbitrary closed line segments in the complex plane. The main result is a geometrical characterization of the functions z→eaz+b, z→(az+b)n and z→(az+b)α+iβ with a,b,∈ℂ, α,β∈ℝ, n∈ℤ

On holomorphic functions with certain extremal properties of its absolute values