【摘 要】
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For a given sequence {zn}n≥0 of positive real numbers, {zn}n≥0 is said to be log-concave (or log-convex) if z2n ≥ zn-1zn+1 (or zn2 ≤ zn-1zn+1) for all n ≥
【机 构】
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DepartmentofMathematics,ShanghaiUniversity,Shanghai200444,China
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For a given sequence {zn}n≥0 of positive real numbers, {zn}n≥0 is said to be log-concave (or log-convex) if z2n ≥ zn-1zn+1 (or zn2 ≤ zn-1zn+1) for all n ≥ 1 and {zn}n≥0is said to be log-balanced if {zn}n≥0 is log-convex and {zn/n!}n≥0 is log-concave.Logconcavity and log-convexity are instrumental in obtaining the growth rate of a sequence and they are also sources of inequalities.It is clear that a sequence {zn}n≥0 is log-convex (log-concave) if and only if its quotient sequence {zn+1/zn}n≥0 is nondecreasing (nonincreasing).In fact, a log-balanced sequence is log-convex, but its quotient sequence does not grow too fast.
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