On a boundary property of analytic functions

Abstract Let f be an analytic function in the unit disc | z | < 1 $|z|<1$ on the complex plane C $\mathbb {C}$ . This paper is devoted to obtaining the correspondence between f ( z ) $f(z)$ and z f ′ ( z ) $zf'(z)$ at the point w, 0 < | w | = R < 1 $0<|w|=R< 1$ , such that | f ( w ) | = min { | f ( z ) | : f ( z ) ∈ ∂ f ( | z | ≤ R ) } $|f(w)|=\min \{|f(z)|: f(z)\in\partial f(|z|\leq R) \}$ . We present several applications of the main result. A part of them improve the previous results of this type