Garrido M. I., Jaramillo J. A.'s A Banach-Stone Theorem for Uniformly Continuous Functions PDF

By Garrido M. I., Jaramillo J. A.

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Let f : I ⊆ R → R be a differentiable convex mapping on ˚ I and a, b ∈˚ I with a < b. 31) f (a) + f (b) 1 − 2 b−a b f (x) dx ≥ max {|A| , |B| , |C|} ≤ 0 a where A := 1 b−a b x− a a+b 1 |f (x)| dx − 2 4 f (b) − f (a) 1 B := + 4 b−a a+b 2 b |f (x)| dx; a b f (x) dx − f (x) dx a+b 2 a and C := 1 b−a b x− a a+b 2 |f (x)| dx. 4. FURTHER INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS Proof. As f is convex on I, the mappings f and x − on [a, b] and we can apply Lemma 1. 32) ≥ max b x− x− a a+b 2 31 are synchronous b a+b 2 dx f (x) dx a where b A¯ := (b − a) x− a b ¯ := (b − a) B b a+b |f (x)| dx − 2 x− a a b a+b f (x) dx − 2 b a+b dx 2 x− |f (x)| dx, a a+b dx 2 x− a b f (x) dx a and b C¯ := (b − a) x− a a+b 2 b |f (x)| dx − x− a b a+b 2 |f (x)| dx.

The following corollary also holds [38]. Corollary 8. With the above assumptions and if the condition that f (a + b − x) = f (x) is satisfied for all x ∈ [a, b] , then we have the inequality: 1 b−a b f (x) dx − f a a+b 2 ≥ 1 b−a b |f (x)| dx − f a a+b 2 ≥ 0. 2. Applications for Special Means. It is well-known that the following inequality holds (G − I − A) G (a, b) ≤ I (a, b) ≤ A (a, b) where, we recall that √ G (a, b) := ab is the geometric mean, bb aa 1 I (a, b) := · e 1 b−a is the identric mean, and a+b 2 is the arithmetic mean of the nonnegative real numbers a < b.

A As it is clear that a+b 2 b b (x − a) f (x) dx + p (x) f (x) dx = a (x − b) f (x) dx, a+b 2 a the required identity is proved. Remark 20. 38) b a+b 2 f (x) dx − f a = 1 b−a b q (x) f (x) dx a where q (x) :=   a − x, x ∈ a, a+b 2  b − x, x ∈ a+b 2 ,b which will be more appropriate, later, for our purposes. The following theorem also holds [34]: Theorem 28. Let f : I ⊆ R → R be a differentiable mapping on ˚ I, a, b ∈˚ I, p with a < b and p > 1. 39) f a+b 2 1 − b−a b a 1 1 (b − a) p f (x) dx ≤ 2 (p + 1) p1 b 1 q q |f (x)| dx a .

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A Banach-Stone Theorem for Uniformly Continuous Functions by Garrido M. I., Jaramillo J. A.


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