Observations on some cardinality bounds

We try to find a common extension of two cardinal inequalities for Lindelöf spaces. Using an estimate of the number of Gδ points due to Balogh, we improve a result of Juhász and Spadaro. A cardinal inequality for linearly Lindelöf Tychonoff spaces proved by Arhangel'skiÄ­ and Buzyakova should be actually true for Hausdorff spaces. We observe this happens under some restrictions on cardinal arithmetics, including a consequence of Martin's axiom. Finally, we address the question to estimate the cardinality of a first countable linearly H-closed space.