MONOTONICITY OF SEQUENCES INVOLVING CONVEX FUNCTION AND SEQUENCE

摘要

Let f be an increasing convex (concave,respectively)funcition defined on [0,1]and {a_i}_i∈N be an increasing positive sequence such that{i(a_i/a_i+1-1)}decreases(i{a_i+1/a_i-1)}_i∈N increases,respectively),then the sequence{1 ∑_i~n=1~f(ai/a_n)}_n∈N is decrasing.let f be an increasing convex (concave,respectively)positive function defined on[0,1]and be an increasing convex positve function defined on [0,∞]such that(0)=0 and the sequence{φ(i)[φ(i)/φ(i+1)-1]}_i∈N decreases, then the sequence {1/φ(n)∑_i=1~n f(φ(i)/φ(n))}_n∈N is decreasin.As applications, taking special sequence {ai}_i∈N and special functions f and φ, many new inequalities between ratios of means are obtained,and the Alzer's inequality,the Minc-Sathre's inequality,and the like,are recovered.