A refinement of Schwarz's lemma at the boundary

UDC 517.5 We investigate a boundary version of the Schwarz lemma for analytic functions. In addition, an analytic function satisfying the equality case is found by deducing inequalities related to the modulus of the derivative of analytic functions at a boundary point of the unit disk. Some coefficients used in the Taylor expansion of the function are considered in these inequalities. In the last theorem, by analyzing the Taylor expansion about two points, we obtain the modulus of the derivative of the function at point 1.