Entire functions having small logarithmic sums over certain discrete subsets (Q1127069)

Let \(h>0\) and let \(\Lambda(\subset\mathbb{R})\) be a relatively \(h\)-dense sequence, i.e., outside a bounded set, any closed interval of length \(h\) contains at least one point of \(\Lambda\). Suppose that the elements of \(\Lambda\) have the extra property of separation: \(\lambda_{n+2} -\lambda_n\geq h\). Then for all entire functions \(f\) of exponential type \(\leq B\leq B_0\) verifying \[ \Sigma \biggr[ \log^+ \bigl| f(\lambda) \bigr| /(\lambda^2+1) \biggr] \leq\eta \quad (\lambda \in \Lambda), \] if \(B_0<T_*/h\) and \(\gamma>0\) \((T_*\approx 0.44)\), there exists \(\eta_0>0\) such that \(\forall\eta\leq\eta_0\), \(\exists C_\eta>0\) and we have \[ \bigl | f(z) \bigr|\leq C_\eta\exp \bigl(B | y|+ \gamma| z| \bigr) \quad (z\in \mathbb{C}). \] This is an extension of one of \textit{P. Koosis}' results [Math. Fiz. Anal. Geom. 2, No. 2, 212-231 (1995; Zbl 0840.30012)].