Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

In 2012, Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form \(\{ \sqrtn]{{z_n }}\} \) , where {z _n } is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {z _n+1/z _n } is increasing (resp., decreasing), then the sequence \(\{ \sqrtn]{{z_n }}\} \) is strictly increasing (resp., decreasing) subject to a certain initial condition. We also give some sufficient conditions when {z _n+1/z _n } is increasing, which is equivalent to the log-convexity of {z _n }. As consequences, a series of conjectures of Zhi-Wei Sun are verified in a unified approach.