Ranges of certain functionals in classes of regular functions

Abstract

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

$$\int_0^{2\pi } {\left| {\operatorname{Re} \left[ {e^{i\lambda } g(z) - \beta \cos \lambda } \right]} \right|} /\left( {1 - \beta } \right)\cos \lambda \left| {d\theta \leqslant \kappa \pi ,} \right.z = re^{i\theta } ,$$

, 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.