On certain subclasses of bounded univalent functions

Let = {z ∈ ℂ; |z| < 1}, T = {z ∈ ℂ; |z|=1}. Denote by S the class of functions f of the form f(z) = z + a₂z² + ... holomorphic and univalent in , and by S(M), M > 1, the subclass of functions f of the family S such that |f(z)| < M in . We introduce (and investigate the basic properties of) the class S(M,m;α), 0 < m ≤ M < ∞, 0 ≤ α ≤ 1, of bounded functions f of the family S for which there exists an open ${\displaystyle arc{I}_{\alpha }=I_{\alpha }\left(f\right)\subset T}$ of length 2πα such that ${\displaystyle {\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le M}$ for every ${\displaystyle z₀\in I_{\alpha }}$ and ${\displaystyle {\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le m}$ for every ${\displaystyle z₀\in T {Ī}_{\alpha }}$.