Convex 0 Journal

Basic short note on the definition of s2-convexity I.M.R. Pinheiro∗ January 26, 2009

Abstract In this little, but fundamental note, we reveal the missing secret about Ks2 : Basically, we have forgotten a few cases included in Convexity when working on its proper extension so far, and have also not thought of the well-posedness rules, as for mathematical definitions, when not spelling out an enthymeme for the definition of Ks2 to make sense with its creational motivations. All the just mentioned problems will be addressed and solved by us here.

AMS 30C45. keywords: convex, S−convex,s2 −convex, s2 -convex, function, s−convex

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Introduction

As seen in [Pinheiro 2008], the determination of the functional class Ks2 is provided by the following definitions: Definition 1. A function f : X− > < is said to be s2 -convex if the inequality f (λx + (1 − λ)y) ≤ λs f (x) + (1 − λ)s f (y) holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; X ⊂ <+ . Remark 1. If the complementary concept is verified, then f is said to be s2 −concave. Therefore, if we write about Convexity, sufficing making s = 1 on the above inequality, we get: Definition 2. A function f : X− > < is said to be convex if the inequality f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; X ⊂ <+ . ∗

RGMIA, e-mail: [email protected], Address: PO BOX 12396, A’BECKETT ST, MELBOURNE, VICTORIA, AUSTRALIA, 8006

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In this paper, we make use of the following sequence of presentation: • Group of generators of counter-examples to the claim that s2 convexity, the way it is found defined so far, extends the concept of Convexity properly; • Proposed fixing in what regards the just mentioned counter-examples; • Conclusion; • References.

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Counter-examples

Suppose f (x) = x2 − 5, and x ∈ [0, 2], what makes our function always negative. We then have the following sequence of contradiction with the definition of the class Ks2 , s 6= 1: f (λx + (1 − λ)y) ≤ λs f (x) + (1 − λ)s f (y)

⇐⇒ (λx + (1 − λ)y)2 − 5 ≤ λs (x2 − 5) + (1 − λ)s (y 2 − 5) what is clearly untrue for at least x = 0, y = 1, λ = 0.5, and s = 0.5. As it is the case that the S-convexity phenomenon breaks the equality of the linearity condition, in favor of the right side of the inequality, unless the function in modulus equates its original value, we will hold an untrue assertion when trying to apply the definition, as seen so far, for S-convexity to convex functions of that sort. Therefore, it can only be the case that the function f (x) = −x, and all functions alike, will only be included in Ks2 if changes are made to its definition. The note, thus, regards observing that no function with non-negative domain, and not entirely null, will satisfy Ks2 pertinence requirements if |f (x)| 6= f (x), on top of fixing the current definition for the S-convexity phenomenon.

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Proposed fixing in the definition of Ks2

Definition 3. A function f : X− > <, where |f (x)| = f (x), is told to belong to Ks2 if the inequality f (λx + (1 − λ)y) ≤ λs f (x) + (1 − λ)s f (y) holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; s : 0 < s ≤ 1; X ⊆ <+ . 2

Definition 4. A function f : X− > <, where |f (x)| = −f (x), is told to belong to Ks2 if the inequality 1

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f (λx + (1 − λ)y) < λ s f (x) + (1 − λ) s f (y) holds ∀λ ∈ [0, 1]; ∀x, y ∈ X; s : 0 < s ≤ 1; X ⊆ <+ . Remark 2. If the complementary concept is verified, then f is said to be s2 −concave. Remark 3. Notice now that, with this small fixing on the definition, we achieve the expected result: Whenever the value of the convex function does not equate its modulus result, we then having a negative image, we get a higher curve for limit by taking less negativity than we had, originally, in the limiting curve when including the function in the group of convex functions.

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Conclusion

In this short note, we have produced needed fixing to the definition of the functional class Ks2 . By now, several problems have ALSO been noticed by us regarding Ks1 and we do intend to produce a note on that definition as well soon. We deal with S−convexity little by little from now onwards, due to both its relevance and controversial material up to 2001, when we started working with it.

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References

[Pinheiro 2008] M. R. Pinheiro. Convexity Secrets. Trafford Publishing. 2008. ISBN: 1425138217.

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