Results are presented from data recorded in 2009 by the PHENIX experiment at the Relativistic Heavy Ion Collider for the double-longitudinal spin asymmetry, $A_{LL}$, for $pi^0$ and $eta$ production in $sqrt{s} = 200$ GeV polarized $p$$+$$p$ collisions. Comparison of the $pi^0$ results with different theory expectations based on fits of other published data showed a preference for small positive values of gluon polarization, $Delta G$, in the proton in the probed Bjorken $x$ range. The effect of adding the new 2009 pz data to a recent global analysis of polarized scattering data is also shown, resulting in a best fit value $Delta G^{[0.05,0.2]}_{mbox{DSSV}} = 0.06^{+0.11}_{-0.15}$ in the range $0.05<x<0.2$, with the uncertainty at $Delta chi^2 = 9$ when considering only statistical experimental uncertainties. Shifting the PHENIX data points by their systematic uncertainty leads to a variation of the best-fit value of $Delta G^{[0.05,0.2]}_{mbox{DSSV}}$ between $0.02$ and $0.12$, demonstrating the need for full treatment of the experimental systematic uncertainties in future global analyses.
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