Ranges of certain functionals in classes of regular functions

LetP_κ,n(λ,β) be the class of functions\(g(z) = 1 + \sum\nolimits_{v = n}^\infty {c_\gamma z^v }\), regular in ¦z¦<1 and satisfying the condition, 0 < r < 1 (κ⩾2,n⩾1, 0⩽Β<1, -π<λ<π/2;M_κ,n(λ,β,α),n⩾2, is the class of functions\(f(z) = z + \sum\nolimits_{v = n}^\infty {a_v z^v }\), regular in¦z¦<1 and such thatF_α(z)∈P_κ,n−1(λ,β), where\(F_\alpha (z) = (1 - \alpha )\frac{{zf'(z)}}{{f(z)}} + \alpha (1 + \frac{{zf'(z)}}{{f'(z)}})\) (0⩽α⩽1). Onr considers the problem regarding the range of the system {g^(v−1)(z_ℓ)/(v−1)!}, ℓ=1,2,...,m,v=1,2,...,N_ℓ, on the classP_κ,1(λ,β). On the classesP_κ,n(λ,β),M_κ,n(λ,β,α) one finds the ranges of C_v, v⩾n, a_m, n⩽m⩽2n-2, and ofg(ς),F_ℝ(ς), 0<¦ξ¦<1, ξ is fixed.