Research On Log-concavity And Log-convexity Of Combinatorial Sequences

Unimodality problem is one of the enduring issues in the study of combinatorial se-quences.Log-concavity and log-convexity are important parts of the unimodality problem.The research on log-concavity and log-convexity can help us to understand the state,distribution,and variation for growth rate of combinatorial sequences.On the other hand,log-concavity and log-convexity are also important sources of combinatorial inequalities.In addition,they play important roles in combinatorial estimation.In this thesis,we study log-concavity and log-convexity of various combinatorial sequences by several methods.We mainly investi-gate log-balancedness of combinatorial sequences(Log-balancedness is a special case of log-convexity).We give some new sufficient conditions for log-balancedness of combinatorial sequences.In addition,we also study log-concavity(log-convex)of some special sequences.The main frames of this thesis are as follows.Chapter 2 of this thesis gives some new sufficient conditions for log-balancedness of combinatorial sequences.Firstly,we show that the product of two log-convex sequences is log-balanced under a mild condition.Based on the result,we not only investigate the log-balancedness of some sequences related to some famous combinatorial sequences,but also dis-cuss the log-balancedness of generalized Motzkin numbers Mn(b,c)and its relavant sequences.Secondly,we study the sufficient conditions for log-balancedness of the sum sequence of two log-balanced sequences.And then,we prove an important property of log-balanced sequences,that is,the arithmetic square root sequence of a log-balanced sequence is still log-balanced.Finally,we apply the definition of log-balancedness to prove log-balancedness of some se-quences.Chapter 3 of this thesis focuses on log-convexity of several special sequences.For the Cauchy numbers of the first kind {an}n≥0 and Cauchy numbers of the second kind {bn}n≥0,we consider the log-convexity of some sequences related to {an}n≥0 and {bn}n≥0.On the other hand,we discuss convexity of generalized Cauchy numbers of the first kind {cf[1](n)} and generalized Cauchy numbers of the second kind[cf[2](n)},where f =(f0,f1,f2,...,fn,...)is a sequence of nonnegative real numbers.We give some sufficient conditions for convexity of{(-1)ncf[1](n)}(or {cf[2](n)}).Chapter 4 of this thesis gives some new sufficient conditions for log-concave of combi-natorial sequences.We mainly study the log-concavity of {znn}n>0,where {Zn}n>0 is a log-concave(log-convex)sequence.In addition,we use the sufficient conditions to prove log-concavity of some sequences.