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We give a generalization of some result of J. Janowski and J. Stankiewicz [2]. Introduction. Let S∗(γ) (|γ| < π/2) denote the class of γ-spiral-starlike functions, i.e., functions F (z) holomorphic in the disc K1 = {z; |z| < 1} and satisfying the conditions F (0) = 0 , F ′(0) = 1 , Re { e−iγ zF ′(z) F (z) } > 0 , z ∈ K1 . Among the many results concerning this class we have the sharp estimation [1] (1) m1(r, γ) ≤ ∣∣∣∣zF ′(z) F (z) ∣∣∣∣ ≤ m2(r, γ) , where m1(r, γ) = √ 1 + 2r2 cos 2γ + r4 − 2r cos γ 1− r2 , (2) m2(r, γ) = √ 1 + 2r2 cos 2γ + r4 + 2r cos γ 1− r2 , (3) r = |z| < 1 , F ∈ S∗(γ) . Let H denote the class of holomorphic functions in K1. Define Ω = {w ∈ H; |w(x)| ≤ 1 for z ∈ K1}. Let f1, f2 be two holomorphic functions in K1. We say that f1 is majorized by f2 in KR = {z; |z| < R} and write f1 f2 if there exists a holomorphic function φ such that |φ(z)| ≤ 1 and f1(z) = φ(z)f2(z), z ∈ KR. J. Janowski and J. Stankiewicz [2] considered the following problem. Let A, B be two fixed classes of holomorphic functions in K1. Find the smallest 1991 Mathematics Subject Classification: Primary 30C45.

On some majorization of derivatives in the class S*(γ)