Representations for the sharp coefficient in an estimate of the modulus of the -th derivative of an analytic function in the unit disk are obtained. It is assumed that the boundary value of the real part of the function on belongs to . The maximum of a bounded factor in the representation of the sharp coefficient is found. Thereby, a pointwise estimate of the modulus of the -th derivative of an analytic function in with a best constant is obtained. The sharp coefficient in the estimate of the modulus of the first derivative in the explicit form is found. This coefficient is represented, for , as the product of monotonic functions of .
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Gershon Kresin, Sharp and maximized real-part estimates for derivatives of analytic functions in the disk. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 24 (2013), no. 1, pp. 95–110DOI 10.4171/RLM/646