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Let f be a strictly increasing smooth function, such that f′ is comparable to a weight α′ which is locally doubling and satisfies a non-triviality condition to be explained in the paper. We construct a meromorphic inner function J, such that f−arg(J) is bounded and arg(J)′ is comparable to α′ up to polynomial loss. We give two applications of this result. The first is a sufficient density conditio
