  In the present investigation, by applying two different normalizations of the
Jackson and Hahn-Exton $q$-Bessel functions tight lower and upper bounds for
the radii of convexity of the same functions are obtained. In addition, it was
shown that these radii obtained are solutions of some transcendental equations.
The known Euler-Rayleigh inequalities are intensively used in the proof of main
results. Also, the Laguerre-P\'olya class of real entire functions plays an
important role in this work.
