Two Inequalities for Convex Functions  

作　　者：PingZhiYUAN HaiBoCHEN 

机构地区：[1]DepartmentofMathematics,SunYat-SenUniversity,Guangzhou510275,P.R.China [2]DepartmentofMathematics,CentralSouthUniversity(TiedaoCampus),Changsha410075,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2005年第1期193-196,共4页数学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(No.10071016)

摘　　要：Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).Let a_0 < a_1 < … < a_n be positive integers with sums Σ_(i=0)~n∈_ia_i(∈_i = 0,1) distinct. P. Erdos conjectured that Σ_(i=0)~n 1/a_i ≤ Σ_(i=0)~n 1/2~i. Thebest known result along this line is that of Chen: Let f be any given convex decreasing function on[A, B] with α_0, α_1, …, α_n , β_0, β_1, …, β_n being real numbers in [A, B] with α_0 ≤α_1 ≤ … ≤ α_n, Σ_(i=0)~n α_i ≥ Σ_(i=0)~n β_i, k = 0, …, n. Then Σ_(i=0)~n f(α_i) ≤Σ_(i=0)~n f(β_i). In this paper, we obtain two generalizations of the above result; each is ofspecial interest in itself. We prove:Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α_0,α_1, …, α_n , β_0, β_1, …, β_n, α'_0, α'_1, …, α'_n , β'_0, β'_1, …, β'_n be realnumbers in [A, B] with α'_0 ≤ α'_1 ≤ … ≤ α_n. Then Σ_(i=0)~n f(α_i)g(α'_i) ≤ Σ_(i=0)~nf(β_i)g(β'_i), k = 0, …, n. Theorem 2 Let f be any given convex decreasing function on [A, B]with k_0, k_1, …, k_n being nonnegative real numbers and α_0, α_1, …, α_n , β_0, β_1, …,β_n being real numbers in [A, B] with α_0 ≤ α_1 ≤ … ≤ α_n, Σ_(i=0)~t k_i α_i ≥ Σ_(i=0)~tk_iβ_i, t = 0, …, n. Then Σ_(i=0)~t k_if(α_i) ≤ Σ_(i=0)~t k_if_(β_i).

关 键 词：Convex functions Finite sums Limits INEQUALITIES 

分 类 号：O174.13[理学—数学]