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18.
Two Inequalities for Convex Functions
Ping Zhi YUAN, Hai Bo CHEN
数学学报(英文版)
2005, 21 (1):
193-196.
DOI: 10.1007/s10114-004-0413-4
Let a0 < a1 < … < an be positive integers with sums εiai( εi= 0,1) distinct. P. Erdös conjectured that 1/ ai≤ 1/2 #em/em#. The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α0, α1, …, αn, β0, β1, …, βn being real numbers in [A, B] with α0 ≤ α1 ≤ … ≤ αn, ai≥ βi,k=0,…,n.Then f(ai)≤ f(βi). 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 α0, α1, …, αn, β0, β1, …, βn, α'0, α'1, …, α'n, β'0, β'1, …, β'n be real numbers in [A, B] with α0 ≤ α1 ≤ … ≤ αn, ai≥ βi,t=0,…,n,α'0 ≤ α'1 ≤ … ≤ α'n, α'i≥ β'#em/em#,t=0,…,n.Then f(ai)g(α'i)≤ f(βi)g(β'#em/em#).
Theorem 2 Let f be any given convex decreasing function on [A, B] with k0, k1, …, kn being nonnegative real numbers and α0, α1, …, αn, β0, β1, …, βn being real numbers in [A, B] with α0 ≤ α1 ≤ … ≤ αn, kiai≥ kiβi,t=0, …,n.Then kif(ai)≤ kif(βi).
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