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REVISIONAL WORKS FOR PREVIOUS RESULTS INVOLVING Ks2 I.M.R. PINHEIRO

Abstract. In this paper, we get to criticize a few results attained by Dragomir et al. in 1999 for the classes Ks2 , and unfortunately nullify them in vast majority. As a side dish, we get to refine our definitions, probably for the last time, now in terms of the supreme we had added as minima domain intervals mark.

1. Introduction So far, our definitions, for the phenomenon of the s2 -convexity, resume to ([Pinheiro 2009]): Definition 1. A function f : X− > <, where |f (x)| = f (x), is told to belong to Ks2 if the inequality f (λx + (1 − λ)(x + δ)) s ≤ λ f (x) + (1 − λ)s f (x + δ) (1) holds ∀λ ∈ [0, 1], ∀x ∈ X, s = s2 /0 < s2 ≤ 1, ∀δ; δ ≥ sup (as − a), X ⊆ <, a ∈ [0, 1]. Definition 2. A function f : X− > <, where |f (x)| = −f (x), is told to belong to Ks2 if the inequality f (λx + (1 − λ)(x + δ)) 1

1

≤ λ s f (x) + (1 − λ) s f (x + δ)

(2) 1

holds ∀λ ∈ [0, 1], ∀x ∈ X, s = s2 /0 < s2 ≤ 1, ∀δ; δ ≥ sup (a − a s ), X ⊆ <, a ∈ [0, 1]. Remark 1. If the inequality is obeyed in the reverse situation by f , then f is told to be s2 −concave. In [Dragomir 1999], we find the following results for the above mentioned group of functions: Theorem 1. Let1 f be a function on [a, b] which is s-convex in the second sense. Then for a < y < z < b we have   f (a) f (b) s , (3) |f (y) − f (z)| ≤ (z − y) max (b − y)s (z − a)s 1991 Mathematics Subject Classification. AMS 30C45. Key words and phrases. convex, S−convex,s1 −convex, dragomir, pearce, function, s−convex. 1Theorem 1.4, p. 688, from [Dragomir 1999]. 1

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so the f is locally Holder continuous of order s on (a, b). Thus, f is Riemann integrable on [a, b]. Theorem 2. Let2 f be a function on [a, b] which is s-convex in the second sense. If f (c) = 0 for some c ∈ [a, b] then f (x) ≤ f (y) if c ≤ x ≤ y ≤ b and f (x) ≥ f (y) if a ≤ x ≤ y ≤ c (4). Let f be Lebesgue integrable on [a, b]. Definition 3.

RbRb F (t) =

a

a

f (tx + (1 − t)y)dxdy (b − a)2

(5)

for3 t ∈ [0, 1]. Theorem 3. Let4 f be s-convex in the second sense on [a, b]. Then F is also 1 1 s-convex in the second sense and F ( 2+t ) = F ( 2−t ) for t ∈ [0, 1] (6). We go through the following presentation sequence: • Criticisms; • Refinement on the definition of s2 -convexity; • Conclusion; • References. 2. Criticisms The worst problems of all seem to originate from the proofs contained in [Dragomir 1999], but the theorems also suffer from some serious problems, which are all found besetting their soundness. In the lines below, we copy the proof in the paper by Dragomir et al. up to the point in which we find impossible that Mathematics supports their claims, abandoning then the rest of the proof and making remarks about it. • In (3), we find t = z−y b−y , which is supposed to become the parameter for the application of the definition of s2 -convexity. However, even though this definition for t does return a value which is certainly less than one, once a < y < z < b in the theorem heading, t will not ever equate 0 or 1. Notwithstanding, if it applies to all values of t ∈ [0, 1], it also applies for those values inside of the interval, so that it is not immediately absurd. However, on top, t should be a constant each time the definition is applied, not a variable in itself. It also cannot be mentioned this way, in the same definition application which uses y, the same y to define t as a variable. The next basic problem is with z = (1 − t)y + tb replacing the left hand side of the definition inequality for s2 −convexity. Notice that b is the upper limit of the domain interval for f in the sequence of reasoning exposed by Dragomir et al. However, the left side of the definition inequality demands a variable to run over the domain slice 2Theorem 1.5, p. 688, from [Dragomir 1999]. 3This definition is found in the page 692, [Dragomir 1999]. 4Theorem 4.1, p. 692, from [Dragomir 1999].

REVISIONAL WORKS FOR PREVIOUS RESULTS INVOLVING Ks2

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under consideration, duty which a constant may not commit to, unless applying the definition for a purpose which is not generalization over it. What follows that is mathematical nonsense, therefore nullifying that result until someone else may come up with a reasonable mathematical proof on it; • In (4), we read, in the proof, that c ≤ x ≤ y ≤ b implies f (x) ≤ ts f (c)+ (1 − t)s f (y) if x = tc + (1 − t)y. Notice that the same mistake, just pointed out by us, has happened here. Once more, we have a constant (c) with variable (y), what does not match the definition inequality for s2 conditions, therefore also nullifying their result unless someone else is able to come up with a mathematically sound proof for it. The problem is the absence of care with the elements being used in the proof. Simple detailed, or accurate, mathematical specifications, would reveal the nonsense: Notice, for instance, that c should be a constant but becomes a variable given how it appears, therefore same sort of problem once more; • In (5), it is all nonsense. Of course that, just for starters, we would have to propose a few restrictions to all: f must be non-negative and [a, b] = [0, 1], so that just the basics of the intentions get secured. However, t is defined as a constant and, therefore, cannot serve the purposes of a linear combination of two variables, as the definition of S-convexity would demand; • In (6), it also seems that all there is nonsense. There is no way we can assert, even imagining we could be replacing t with each and every member of the interval [0, 1], that F has got the same nature as f , for it is impossible to prove anything without having to impose further restrictions to f , for a single definition application for Ks2 will demand at least three domain members to be involved in the same inequality, what would be impossible to do given what we know about f . It is true, however, that F (0.5 + t) = F (0.5 − t), trivially. 3. Refinement on the definition of s2 −convexity • sup (as − a): Simple reasoning would lead us to the obvious value for a so that the supreme for this subtraction would be reached. Basically, while a lives in the straight line world, as lives above that world, in a curved shape, as we have proved in [Pinheiro2 2009]. Therefore, the maximum distance will have to be achieved, if we are drawing a ninety degree line from the straight line, when a = 0.5, that is, when the maximum for the curved line function is reached, as also proven there; 1 1 • sup (a − a s ): Once more, a lives in the straight line world but a s lives in the curved line world, and the maximum distance is again achieved at a = 0.5. 4. Conclusion In this paper, we have managed to nullify three major results by Dragomir et al. regarding s2 −convexity, believing to be meaningfully contributing to the progress of Mathematics in general in an actual way.

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We also got to produce what seems to be our last refinement in the definition of the s2 −convexity phenomenon: Definition 4. A function f : X− > <, where |f (x)| = f (x), is told to belong to Ks2 if the inequality f (λx + (1 − λ)(x + δ)) s ≤ λ f (x) + (1 − λ)s f (x + δ) (1) holds ∀λ ∈ [0, 1], ∀x ∈ X, s = s2 /0 < s2 ≤ 1, ∀δ; (sup X − x) ≥ δ ≥ (0.5s − 0.5), X ⊆ <. Definition 5. A function f : X− > <, where |f (x)| = −f (x), is told to belong to Ks2 if the inequality f (λx + (1 − λ)(x + δ)) 1

1

≤ λ s f (x) + (1 − λ) s f (x + δ) (2) holds ∀λ ∈ [0, 1], ∀x ∈ X, s = s2 /0 < s2 ≤ 1, ∀δ; (sup X − x) ≥ δ ≥ (0.5 − 1 0.5 s ), X ⊆ <. 5. References [Dragomir 1999] S.S. Dragomir, S. Fitzpatrick. The Hadamard Inequalities For s-convex Functions in the Second Sense. Demonstratio Mathematica. Vol. XXXII, No. 4, 1999. [Pinheiro 2009] M. R. Pinheiro. Minima Domain Intervals and the S-convexity, as well as the Convexity, Phenomenon. Submitted, 2009. [Pinheiro2 2009] M. R. Pinheiro. Short note on the definition of s2 -convexity. Submitted, 2009. PO BOX 12396, A’BECKETT ST, MELBOURNE, VICTORIA, AUSTRALIA, 8006 E-mail address: [email protected]