Abstract
In 2012, Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form \(\{ \sqrt[n]{{z_n }}\} \), 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 \(\{ \sqrt[n]{{z_n }}\} \) 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.
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Wang, Y., Zhu, B. Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences. Sci. China Math. 57, 2429–2435 (2014). https://doi.org/10.1007/s11425-014-4851-x
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DOI: https://doi.org/10.1007/s11425-014-4851-x