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Given an admissible weight w and 0<p<∞, the estimate ∫p|f(z)|^pw(z)dm(z)-|f(0)|^p+∫p|f'(z)|^pψ^p(z)w(z)dm(z) is valid for all holomorphic functions f in the unit disc D. Here,ψ(r)=∫r^1w(t)dt/w(r) is the distortion of w. As an application of the