Some Researches On Log-convexity Of Sequences

Log-concavity(log-convexity),as an important part of unimodality problem,has always been one of the enduring issues in the research on combinatorial sequences.The study of log-concavity(log-convexity)is of great significance.It can help us understand the state,dis-tribution,and variation for growth rate of combinatorial sequences.Furthermore,it is also an essential source of combinatorial inequalities,and has widely applications in combinatorial estimation.In this thesis,we mainly investigate log-convexity of various sequences.In partic-ular,we investigate log-balancedness of sequences.We obtain some new sufficient conditions for log-balancedness of sequences.Moreover,we discuss the log-concavity and log-convexity of certain sequences involving Cauchy numbers of the first kind {an}n?0 and the second kind{bn}n?0.The main results of this thesis are as follows:Chapter 2 of this thesis gives several new sufficient conditions for log-balancedness of sequences.For a log-convex sequence {Zn}n?0,we show that {(?)}n?0 is log-balanced under suitable conditions,where r>0 is a fixed number.On the other hand,suppose that a and b are positive real numbers with b<a and {Zn}n?0 is a log-convex sequence.If the sequence {(?)}n?0 is log-balanced,then so is the sequence {Znb}n?0.Using these new results,we investigate the log-balancedness of a number of log-convex sequences(These sequences are related to many famous combinatorial numbers).For example,for derangement numbers dn,we prove that the sequence {(?)}n?3 is log-balanced when r>2.In addition,we discuss the log-convexity of a class of linear recurrence sequences.Chapter 3 of this thesis deals with the log-balancedness of the sequence {(?)}n?1.We obtain several new sufficient conditions for log-balancedness of {(?)}n?1,where {zn}n?0 is log-concave.As applications,we derive the log-balancedness of a number of sequences related to Catalan numbers,derangement numbers,Motzkin numbers,Fine numbers,central Delannoy numbers,and so on.Chapter 4 of this thesis focuses on the log-convexity of some sequences related to Cauchy numbers of the first kind {an}n?0 and the second kind {bn}n?0.For example,we prove that sequences {(?)}n?2 and {(?)}n?1 are both log-balanced.