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Two Inequalities for Convex Functions

Authors:	Ping Zhi Yuan Hai Bo Chen

Institution:	(1) Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, P. R. China;(2) Department of Mathematics, Central South University (Tiedao Campus), Changsha 410075, P. R. China

Abstract:	Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums ${\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} }$ distinct. P. Erd?s conjectured that ${\sum\nolimits_{i = 0}^n {1/a_{i} \leqslant {\sum\nolimits_{i = 0}^n {1/2^{i} } }} }.$ The best 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, ${\sum\nolimits_{i = 0}^k {\alpha _{i} \geqslant {\sum\nolimits_{i = 0}^k {\beta _{i} ,k = 0, \ldots ,n} }} }.$ Then ${\sum\nolimits_{i = 0}^n {f{\left( {\alpha _{i} } \right)} \leqslant {\sum\nolimits_{i = 0}^n {f{\left( {\beta _{i} } \right)}} }} }.$ In this paper, we obtain two generalizations of the above result; each is of special 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 real numbers in A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n, ${\sum\nolimits_{i = 0}^t {\alpha _{i} \geqslant {\sum\nolimits_{i = 0}^t {\beta _{i} ,t = 0, \ldots ,n,} }} }$ α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n, ${\sum\nolimits_{i = 0}^t {{\alpha }'_{i} \geqslant {\sum\nolimits_{i = 0}^t {{\beta }'_{i} ,t = 0, \ldots ,n.} }} }$ Then ${\sum\nolimits_{i = 0}^n {f{\left( {\alpha _{i} } \right)}g{\left( {{\alpha }'_{i} } \right)} \leqslant {\sum\nolimits_{i = 0}^n {f{\left( {\beta _{i} } \right)}g{\left( {{\beta }'_{i} } \right)}.} }} }$ 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, ${\sum\nolimits_{i = 0}^t {k_{i} \alpha _{i} \geqslant {\sum\nolimits_{i = 0}^t {k_{i} \beta _{i} ,t = 0, \ldots ,n.} }} }$ Then ${\sum\nolimits_{i = 0}^n {k_{i} f{\left( {\alpha _{i} } \right)} \leqslant {\sum\nolimits_{i = 0}^n {k_{i} f{\left( {\beta _{i} } \right)}.} }} }$

Keywords:	Convex functions Finite sums Limits Inequalities
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