Denote by S the class of functions f(z) a_ν z~ν regular and univalent in the unit circle |z|<1, by S the subclass of S each function w=f(z) of which maps |z|<1 on a domain D_f starshaped with respect to w=0, and finally by S the subclass of S,whose function f(z) are characterized by the condition The class K is the subclass of S such that the image D_f obtained by w=f(z) of K is convexWe write σ_n(z)=z+a_2z~2+…+a_nz~n, the sections of the expansion f(z)Rachmanof proves that, if f(z) ∈ K, then all the sections σ_n (z) except σ_4 (z), are Univalent in the circle |z| <1/2. In the present paper, we settle Rachmanof's problem on σ_4(z). Indeed, what is more, we can prove the followingTheorem A. If f(z) a_νz~ν∈S,then 2σ_n(1/2 z)∈S,n=1,2,3…, and 2σ_n(1/2 z)∈S, n=1, 2, , , , 6, …. Moreover, 3σ_n(1/3 z)∈S, provided that n≠3. Indeed, 4σ_n (1/4 z)∈K; furthermore, if f(z) ∈ S, then 4σ_n(1/4 z)∈S, n 1, 2, 3, …The factors 1/2,1/3,1/4,in the respective cases are not allowed to be increased.The proof is based chiefly on the principle of subordination. In fact, we can establishTheorem B. If f(z)∈S, then and there exists an increasing function a(θ) with a(θ) = a(θ+0) (0≤θ<2π), a(2π) - a(0) = 2π, such that and that This last formula of representation implies f(z)∈S.Hence we can deduce the following precise estimates): z = re~(iθ), 0 ≤ r < 1, All the signs of equality in these relations hold good when and only when f(z)≡f(z)=z/1-ηz,|η| =1.Furthermore, by a famous theorem of C. Caratheodory, we can prove the followingTheorem C. Suppose f(z) a_ν z~ν ∈S. For any integer m ≥2, consider (a_2, a_3,…, a_m) as a point P_m in the 2_(m-2) dimensional real Euclidian space. Then P_m falls into the smallest convex body V_m, containing the closed curve Γ_m, which is defined parametrically by (e~(-iθ), e~(-i2θ),…, e~(-i(m-1)θ)) ,m ≥ 2, 0 ≤ θ ≤ 2π. The case P_m∈Γ_m realizes if and only if f(z)≡f(z)=z(1-zη)~(-1), |η| = 1.Finally, we give some relations ,between a_2 and a_3.Theorem D. if f(z)=z+a_2 z~2+a_3 z~3+…∈S, then the equality holds only for the function f(z)=z[1-ze~(-i arc cos(±1/4))]~(-1); and |a_3-1/2 a_2~2|≤1/2, the equality holds only for the function f(z)=z(1-ηz)~(-λ) (1+ηz)~(-(1-λ)),|η| =1, 0≤λ≤1.
