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We describe the growth of the naturally defined argument of a bounded analytic function in the unit disk in terms of the complete measure introduced by A.Grishin. As a consequence, we characterize the local behavior of a logarithm of an analytic function. We also find necessary and sufficient conditions for closeness of f(z), f∈ H^∞, and the local concentration of the zeros of f.arXiv:0906.1138v1 fatcat:fpwcxns3bnh4ff6t542ti3e5ai