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 {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） s...

Full description

Saved in:
Bibliographic Details
Published inScience China. Mathematics Vol. 57; no. 11; pp. 2429 - 2435
Main Authors Wang, Yi, Zhu, BaoXuan
Format Journal Article
LanguageEnglish
Published Heidelberg Science China Press 01.11.2014
Subjects
Applications of Mathematics
China
Combinatorial analysis
Equivalence
Initial conditions
Mathematical analysis
Mathematics
Mathematics and Statistics
Sequences
Sun
Texts
Zinc
严格递增
充分条件
单调性
数论
猜想
组合序列
证明
锌
log-concavity
11B83
05A10
05A20
monotonicity
log-convexity
sequences
Online AccessGet full text
ISSN1674-7283
1869-1862
DOI10.1007/s11425-014-4851-x

Cover

More Information
Summary: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.
Bibliography:sequences; monotonicity; log-convexity; log-concavity
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.
11-1787/N
ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 23
ISSN:1674-7283
1869-1862
DOI:10.1007/s11425-014-4851-x