Growth estimates for the maximal term and central exponent of the derivative of a Dirichlet series - Fedynyak, S.I.; Filevych, P.V. (Carpathian Mathematical Publications, 2020)
Let A(-∞,+∞], Φ:[a,A)→R be a continuous function such that xσ-Φ(σ)→-∞ as σA for every xR, Φ(x)={xσ -Φ(σ):σ[a,A)} be the Young-conjugate function of Φ, Φ(x)=Φ(x)/x and Γ(x)=(Φ(x)-x)/x for all sufficiently large x, (λ_n) be a nonnegative sequence increasing to +∞, and F(s)=_n=0^∞ a_ne^sλ_n be a Dirichlet series such that its maximal term μ(σ,F)={|a_n|e^σλ_n:n0} and central index u(σ,F)={n0:|a_n|e^σλ_n=μ(σ,F)} are defined for all σ<A. It is proved that if μ(σ,F)(1+o(1))Φ(σ) as σA, then the inequalities _σAμ(σ,F')μ(σ,F)Φ ^-1(σ)1,_σAλ_ u(σ,F') Γ^-1(σ)1, hold, and these inequalities are sharp.