R. B. Paris, SIAM J. Math. Anal. 15, 203-205 (1984; Zbl 0536.33006) derived upper and lower bounds for the ratio by using recurrence relations and Sonine’s integral representation. The present author uses similar techniques to extend such results to general cylinder functions and to improve the results of Paris in the case . Some examples of the author’s results run as follows:
where is the smallest positive zero of .