We report a Giant Metrewave Radio Telescope (GMRT) survey for associated HI 21-cm absorption from 50 active galactic nuclei (AGNs), at $z approx 0.04 - 3.01$, selected from the Caltech-Jodrell Bank Flat-spectrum (CJF) sample. Clean spectra were obtained towards 40 sources, yielding two new absorption detections, at $z = 0.229$ towards TXS 0003+380 and $z = 0.333$ towards TXS 1456+375, besides confirming an earlier detection, at $z = 1.277$ towards TXS 1543+480. There are 92 CJF sources, at $0.01 lesssim z lesssim 3.6$, with searches for associated HI 21-cm absorption, by far the largest uniformly-selected AGN sample with searches for such absorption. We find weak ($approx 2sigma$) evidence for a lower detection rate of HI 21-cm absorption at high redshifts, with detection rates of $28^{+10}_{-8}$% and $7^{+6}_{-4}$% in the low-$z$ ($z < z_{rm med}$) and high-$z$ ($z > z_{rm med}$) sub-samples, respectively. We use two-sample tests to find that the strength of the HI 21-cm absorption in the AGNs of our sample depends on both redshift and AGN luminosity, with a lower detection rate and weaker absorption at high redshifts and high ultraviolet/radio AGN luminosities. Unfortunately, the luminosity bias in our sample, with high-luminosity AGNs arising at high redshifts, implies that it is not currently possible to identify whether redshift evolution or AGN luminosity is the primary cause of the weaker absorption in high-$z$, high-luminosity AGNs. We find that the strength of HI 21-cm absorption does not depend on AGN colour, suggesting that dust extinction is not the main cause of reddening in the CJF sample.
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