A conjecture on monotonicity of a ratio of Kummer hypergeometric functions

Abstract: In the preprint of 1993 the author formulated some conjectures on monotonicity of ratios for exponential series remainders. They are equivalent to conjectures on monotonicity of a ratio of Kummer hypergeometric functions and presumably not proved still. In this short note the most interesting conjecture from the preprint of 1993 is reproduced with its generalizations. In the updated version an important information is added that conjectures 1, 2 and also problems 1, 2 were recently proved by Khaled Mehrez from IPEIM, Tunisia, for details cf. arXiv:1410.6120. Also the results on monotonicity were generalized to basic hypergeometric functions.