The extremely energetic Fermi GRBs 080916C, with its Eiso of ~ 10^{55} erg in 1 keV - 10 GeV and intense GeV emission, and 090323 give us a unique opportunity to test the reliability and extension of spectral energy correlations. Based on Konus/WIND and Fermi spectral measurements, we find that both events are fully consistent with the updated (95 events as of April 2009) Ep,i - Eiso correlation, thus further confirming and extending it and pointing against a possible flattening or increased dispersion at very high energies. This also suggests that the physics behind the emission of peculiarly bright and hard GRBs is the same as for softer and weaker ones. In addition, we find that the normalization of the correlation obtained by considering these two GRBs and the other long ones for which Ep,i was measured with high accuracy by the Fermi/GBM are fully consistent with those obtained by other instruments (e.g., BeppoSAX, Swift, Konus-WIND), thus indicating that the correlation is not affected significantly by detectors limited thresholds and energy bands. Prompted by the extension of the spectrum of GRB 080916C up to several GeVs without any excess or cut-off, we also investigated if the evaluation of Eiso in the commonly adopted 1 keV - 10 MeV energy band may bias the Ep,i - Eiso correlation contributing to its scatter. By computing Eiso from 1 keV to 10 GeV, the slope of the correlation becomes slightly flatter, while its dispersion does not change significantly. Finally, we find that GRB 080916C is also consistent with most of the other spectral energy correlations derived from it, with the possible exception of the Ep,i - Eiso - tb correlation.

9 pages, 5 figures, final revised version accepted for pubblication in Astronomy & Astrophysics (main Journal)