The Schwarz Inequality on the Boundary for Functions Regular in the Disk

By V. Dubinin. 2004

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A refined Schwarz inequality on the boundary (2013, background): “6) are generalized in the paper [5], where j f 0(c)j is estimated according to zeros and the first nonzero Taylor coefficient of f about z1⁄4 0.”
Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros (2007, background): “In (2) for a real z and in the left-hand-side inequality in (3) for b = 1, equality occurs for the function f(z )= z(z + a)/(1 + az), 0 ≤ a ≤ 1. As a supplement to (3), we recall the following result [ 3, Theorem 2 ]. Let a function f be regular in U , |f| < 1i nU , and let this function and its der”