On the analog of the Sălăgean class for Dirichlet series and the solutions of one linear differential equation with exponential coefficients - Sheremeta, M.; Mulyava, O.; Medvedev, M. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.537 In his study of the geometric properties of functions analytic in a disk D = {z|z|<1}, G. S. Sălăgean introduced a class S_j(α) of functions f(z) = z + _k = 2^∞f_kz^k such that Re D^j + 1f(z)D^jf(z) > α[0, 1) for each zD, where D^jf is the Sălăgean derivative. For Dirichlet series F(s) = e^s-_k = 1 ^∞f_k{sλ_k} with f_k0 absolutely convergent in the half plane Π_0 = {sRe s<0}, an analog of the Sălăgean class is the classD_j(α) defined by the condition Re F^(j + 1)(s)F^(j)(s) > α for each sΠ_0. By analogy with the neighborhood of an analytic function in D defined by A. V. Goodman, for FD_j(α), we introduce the concept of a neighborhood O_j,δ(F) and establish the conditions under which all functions from O_j,δ(F) belong to D_j(α_1), 0α_1<α<1, and vice versa. The problem of belonging of solutions of the differential equation d^2wds^2 + (γ_0e^2s + γ_1 e^s + γ_2)w = 0 with real parameters to the class D_j(α) is investigated.