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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 $arc{I}_{\alpha }=I_{\alpha }\left(f\right)\subset T$ of length 2πα such that ${\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le M$ for every $z₀\in I_{\alpha }$ and ${\overline{lim}}_{z\to z₀,z\in }\left|f\left(z\right)\right|\le m$ for every $z₀\in T{Ī}_{\alpha }$.