Schwarz lemma and its applications on the unit disc

Abstract

In this article, a refinement of the Schwarz lemma (boundary Schwarz lemma) is presented for a different class. For the analytic function p(z) = z+b2z2+b3z3+..., defined in the unit disc U satisfying ℜ(p(rz)−p(sz)/(r−s)z)≥0 for z ∈ U, where r, s ∈ C, r ̸= s, |r| ≤ 1, |s| ≤ 1, we estimate a module of angular derivative at the boundary point 1with p(r) = p(s). The sharpness of these estimates is also proved.