Let $f$ be a function regular in $\Re z \geq 0$ whose indicator function satisfies: $$h(\theta) = \limsup_{r \to \infty} \frac{\log |f(r e^{i \theta})|}{r} \leq c < \pi$$ in this half plane. Let $(\lambda_n)_{n \geq 0}$ be an increasing sequence of positive real numbers satisfying $|\lambda_n - n| < L$ and $\lambda_{n + 1} - \lambda_{n} > \delta$ > 0. My question is about the equivalence of the sum of $|f|$ over this sequence and its integral over the positive real axis, more precisely, whether: $$ \sum_{n = 0}^{\infty} |f(\lambda_n)| < \infty\tag{1}\label{eq:series} $$ implies $$ \int_0^{\infty} |f(x)| \, dx < \infty,\tag{2}\label{eq:integral} $$ possibly under any extra hypotheses.
For context: results by Boas and Kjellberg imply that the converse is true, namely \eqref{eq:integral} implies \eqref{eq:series} (they actually prove a result about more general convex functions $\phi$ of $|f|$). Another result by Boas says that \eqref{eq:series} implies \eqref{eq:integral} if we consider instead the entire real axis and a sequence $(\lambda_n)_{n \in \mathbb{Z}}$, i.e.: $$ \sum_{n = -\infty}^{\infty} |f(\lambda_n)| < \infty. $$ implies $$ \int_{-\infty}^{\infty} |f(x)| \, dx < \infty $$ Can this result be extended to the real positive half-axis?
