We investigate the thermal history of the intergalactic medium (IGM) in the redshift interval z = 1.7–3.2 by studying the small-scale fluctuations in the Lyman α forest transmitted flux. We apply a wavelet filtering technique to 18 high-resolution quasar spectra obtained with the Ultraviolet and Visual Echelle Spectrograph, and compare these data to synthetic spectra drawn from a suite of hydrodynamical simulations in which the IGM thermal state and cosmological parameters are varied. From the wavelet analysis we obtain estimates of the IGM thermal state that are in good agreement with other recent, independent wavelet-based measurements. We also perform a reanalysis of the same data set using the Lyman α forest flux probability distribution function (PDF), which has previously been used to measure the IGM temperature–density relation. This provides an important consistency test for measurements of the IGM thermal state, as it enables a direct comparison of the constraints obtained using these two different methodologies. We find the constraints obtained from wavelets and the flux PDF are formally consistent with each other, although in agreement with previous studies, the flux PDF constraints favour an isothermal or inverted IGM temperature–density relation. We also perform a joint analysis by combining our wavelet and flux PDF measurements, constraining the IGM thermal state at z = 2.1 to have a temperature at mean density of T0/[103 K] = 17.3 ± 1.9 and a power-law temperature–density relation exponent γ = 1.1 ± 0.1 (1σ). Our results are consistent with previous observations that indicate there may be additional sources of heating in the IGM at z < 4.