  Fermi-LAT analyses show that the gamma-ray photon spectral indices
Gamma_gamma of a large sample of blazars correlate with the vFv peak
synchrotron frequency v_s according to the relation Gamma_gamma = d - k log
v_s. The same function, with different constants d and k, also describes the
relationship between Gamma_gamma and peak Compton frequency v_C. This behavior
is derived analytically using an equipartition blazar model with a log-parabola
description of the electron energy distribution (EED). In the Thomson regime, k
= k_EC = 3b/4 for external Compton processes and k = k_SSC = 9b/16 for
synchrotron self-Compton (SSC) processes, where b is the log-parabola width
parameter of the EED. The BL Lac object Mrk 501 is fit with a synchrotron/SSC
model given by the log-parabola EED, and is best fit away from equipartition.
Corrections are made to the spectral-index diagrams for a low-energy power-law
EED and departures from equipartition, as constrained by absolute jet power.
Analytic expressions are compared with numerical values derived from
self-Compton and external Compton scattered gamma-ray spectra from Ly alpha
broad-line region and IR target photons. The Gamma_gamma vs. v_s behavior in
the model depends strongly on b, with progressively and predictably weaker
dependences on gamma-ray detection range, variability time, and isotropic
gamma-ray luminosity. Implications for blazar unification and blazars as
ultra-high energy cosmic-ray sources are presented. Arguments by Ghisellini et
al. (2014) that the jet power exceeds the accretion luminosity depend on the
doubtful assumption that we are viewing at the Doppler angle.
