Bernstein-type inequalities for analytic functions represented by power series - Fedynyak, S. I.; Filevych, P. V. (Matematychni Studii, 2024)
Let R(0,+∞] and D_R={zC|z|<R}. Denote by A_R the class of all functions f analytic in D_R such that f(z) ot0. For any function fA_R, let M(r,f)={|f(z)||z|=r} be the maximum modulus, K (r,f)=rM(r,f’)/M(r,f), and μ(r,f)={|a_n(f)|r^nn0 } be the maximal term of the Maclaurin series of the function f, where a_n(f) denotes the n-th coefficient of this series. Suppose that Φ is a continuous function on [a,R) such that for every xR we have xσ-Φ(σ )→-∞ as σR, and let Φ(x)={xσ -Φ(σ)σD_Φ} be the Young conjugate function of Φ, φ(x)=Φ’_+(x) for all x R, and Γ(x)=(Φ(x)-x)/x for all sufficiently large x. Put Δ=_x→+∞xΦ(φ(x)), t(f)=_rRμ(r,f)Φ(r), k(f)=_rR K(r,f)Φ \!^-1(r), k_1(f)=_rR K(r,f)Γ^-1(r), where fA_R. We prove the following results: (a) for any function fA_R such that t(f)1, the inequality k_1(f)1 holds; (b) for an arbitrary positive sequence (r_n) increasing to R, there exists a function fA_R such that t(f)=1 and _n → +∞K(r_n,f)/Γ^-1(r_n)=1; (c) for any function fA_R such that t(f)1, the inequality k(f)1+Δ holds; (d) there exists a function fA_R such that t(f)=1 and k(f)=1+Δ.