On the Young conjugate functions and the behavior of the maximal terms of the derivatives of Dirichlet series (Q5951070)

Let \(F(s)=\sum ^{\infty}_{n=1} a_n \exp \{s \lambda _n \}\), \(s=\sigma +it,\) be a Dirichlet series with \(1\leq \lambda _1 < \lambda _n \uparrow \infty \), \(\liminf _{n \rightarrow \infty } (1/\lambda _n) \log (1/ |a_n|) = A \in (- \infty, \infty ]\), let \(\mu (\sigma, F)= \max \{|a_n |\exp \{\sigma \lambda _n \}: n \geq 1 \}\) be its maximal term and \(P(t)= \log |a_n|\) for \(t=\lambda _n\) and \(P(t)=-\infty \) for \(t \in (0, \infty) \setminus \{\lambda _n \}\). Let \(\Phi \) be a convex function subject to some regularity conditions stated in the paper. The authors prove that \[ \log \mu (\sigma) = O(\Phi (\sigma)), \quad \sigma \uparrow A \] if and only if there exists \(N\in \mathbb{Z}_{+}\) such that \[ \frac{\mu (\sigma, F^{(n)})}{n! (\Phi' (\sigma))^n} \leq \max \Biggl\{ \frac{\mu ( \sigma, F^{(k)})}{k! (\Phi' (\sigma))^k}: 0\leq k \leq N \Biggr\} \] for all \(\sigma < A\) and \(n \in \mathbb{Z}_+\). This theorem is derived from the following property of the Young conjugate functions \(Q\) applied to \(Q(\sigma) = \sup \{ P(t) + \sigma t: t>0 \}\): \[ Q(\sigma) = O(\Phi (\sigma)), \quad \sigma \uparrow A \] if and only if there exists \(B>0\) such that for all \(\sigma <A\) and \(\alpha \geq 0\) \[ Q(\alpha, \sigma) -\log \Gamma (1+ \alpha)-\alpha \log \Phi' (\sigma) \leq \max \{ Q(\beta, \sigma)-\log \Gamma (1+ \beta)-\beta \log \Phi' (\sigma): 0 \leq \beta \leq B \}, \] where \(Q(\alpha, \sigma) = \sup \{ P(t) + \alpha \log t + \sigma t: t \geq 1 \}\).