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

作　　者：WANG Yi ZHU BaoXuan 

机构地区：[1]School of Mathematical Sciences,Dalian University of Technology [2]School of Mathematical Sciences,Jiangsu Normal University

出　　处：《Science China Mathematics》2014年第11期2429-2435,共7页中国科学：数学（英文版）

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)

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.

关 键 词：SEQUENCES MONOTONICITY LOG-CONVEXITY LOG-CONCAVITY 

分 类 号：O157[理学—数学] O174[理学—基础数学]