摘要
In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.
In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn＋1/zn}is increasing（resp.,decreasing）,then the sequence {n√zn} is strictly increasing（resp.,decreasing） subject to a certain initial condition. We also give some sufficient conditions when {zn＋1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.
基金
supported by National Natural Science Foundation of China(Grant Nos.11071030,11201191 and 11371078)
Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20110041110039)
National Science Foundation of Jiangsu Higher Education Institutions(GrantNo.12KJB110005)
the Priority Academic Program Development of Jiangsu Higher Education Institutions(Grant No.11XLR30)
