On approximately monotone and approximately Hölder functions

Goswami, Angshuman R.

doi:10.1007/s10998-020-00351-0

Cited by 6 publications

(13 citation statements)

References 25 publications

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“…It is clear that absolutely subadditive functions are automatically subadditive. On the other hand, as we have proved it in [4], increasingness and subadditivity imply absolute subadditivity. In [4], we also established a formula for the lower and for the upper Φ-monotone and Φ-Hölder envelopes.…”

Section: Introductionmentioning

confidence: 54%

“…On the other hand, as we have proved it in [4], increasingness and subadditivity imply absolute subadditivity. In [4], we also established a formula for the lower and for the upper Φ-monotone and Φ-Hölder envelopes. Furthermore, we introduced a generalization of the classical notion of total variation and proved an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.…”

Section: Introductionmentioning

confidence: 54%

“…It turns out (cf. [4,Proposition 3.2]), that if Φ is of the form Φ(t) = ct p , where c ≥ 0 and p ∈ ]0, 1], then Φ is a subadditive and nondecreasing error function.…”

Section: Characterization Of φ-Monotone Functionsmentioning

confidence: 99%

“…The proof of the last two implications is based on the result of our previous paper [4] which states that the family of Φ-monotone functions is closed under pointwise supremum and infimum.…”

Section: Characterization Of φ-Monotone Functionsmentioning

confidence: 99%

“…[1,2,3,5,6,7]) and have applications in nonsmooth and convex analysis and optimization theory, and also in the theory of functional equations and inequalities. Motivated by the applicability of such concepts, in our former paper [4], we introduced…”

Section: Introductionmentioning

confidence: 99%

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Characterization of approximately monotone and approximately Hölder functions

Goswami¹

2021

Mathematical Inequalities &Amp; Applications

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A real valued function f defined on a real open interval I is called Φ-monotone if, for all x,y ∈ I with x y it satisfies f (x) f (y) + Φ(y − x), where Φ : [0, (I)[→ R + is a given nonnegative error function, where (I) denotes the length of the interval I. If f and − f are simultaneously Φ-monotone, then f is said to be a Φ-Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct Φmonotone and Φ-Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski-and Hermite-Hadamardtype inequalities from the Φ-monotonicity and Φ-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.

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Section: Introductionmentioning

confidence: 54%

Section: Introductionmentioning

confidence: 54%

“…It turns out (cf. [4,Proposition 3.2]), that if Φ is of the form Φ(t) = ct p , where c ≥ 0 and p ∈ ]0, 1], then Φ is a subadditive and nondecreasing error function.…”

Section: Characterization Of φ-Monotone Functionsmentioning

confidence: 99%

Section: Characterization Of φ-Monotone Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterization of approximately monotone and approximately Hölder functions

Goswami¹

2021

Mathematical Inequalities &Amp; Applications

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Generalizing the concept of bounded variation

Goswami

2024

Aequat. Math.

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Let $$[a,b]\subseteq \mathbb {R}$$ [ a , b ] ⊆ R be a non-empty and non singleton closed interval and $$P=\{a=x_0<\cdots <x_n=b\}$$ P = { a = x 0 < ⋯ < x n = b } is a partition of it. Then $$f:I\rightarrow \mathbb {R}$$ f : I → R is said to be a function of r-bounded variation, if the expression $$\sum \nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r}$$ ∑ i = 1 n | f ( x i ) - f ( x i - 1 ) | r is bounded for all possible partitions like P. One of the main results of the paper deals with the generalization of the classical Jordan decomposition theorem. We establish that for $$r\in ]0,1]$$ r ∈ ] 0 , 1 ] , a function of r-bounded variation can be written as the difference of two monotone functions. While for $$r>1$$ r > 1 , under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for $$0<r_1<r_2$$ 0 < r 1 < r 2 , the function class of $$r_1$$ r 1 -bounded variation is contained in the class of functions satisfying $$r_2$$ r 2 -bounded variations. We go through approximately monotone functions and present a possible decomposition for $$f:I(\subseteq \mathbb {R}_+)\rightarrow \mathbb {R}$$ f : I ( ⊆ R + ) → R satisfying the functional inequality $$f(x)\le f(x)+(y-x)^{p}\quad (x,y\in I \text{ with } x<y \text{ and } p\in ]0,1[ ).$$ f ( x ) ≤ f ( x ) + ( y - x ) p ( x , y ∈ I with x < y and p ∈ ] 0 , 1 [ ) . A generalized structural study has also been done in that specific section. On the other hand, for $$\ell [a,b]\ge d$$ ℓ [ a , b ] ≥ d , a function satisfying the following monotonic condition under the given assumption will be termed as d-periodically increasing $$f(x)\le f(y)\quad \text{ for } \text{ all }\quad x,y\in I\quad \text{ with }\quad y-x\ge d.$$ f ( x ) ≤ f ( y ) for all x , y ∈ I with y - x ≥ d . We establish that in a compact interval any function satisfying d-bounded variation can be decomposed as the difference of a monotone and a d-periodically increasing function. The core details related to past results, motivation, structure of each and every section are thoroughly discussed below.

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Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

Barić¹

2022

Mathematics

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As convexity plays an important role in many aspects of mathematical programming, e.g., for obtaining sufficient optimality conditions and in duality theorems, and one of the most important inequalities for convex functions is the Hermite–Hadamard inequality, the importance of this paper lies in providing some new improvements for convex functions and new directions in studying new variants of the Hermite–Hadamard inequality. The first part of the article includes some known concepts regarding convex functions and related inequalities. In the second part of the study, a derivation of the Hermite–Hadamard inequality for convex functions of higher order is given, emphasizing the purpose and importance of some quadrature formulas. In the third section, the applications of the main results are presented by obtaining Hermite–Hadamard-type estimates for various classical quadrature formulas such as the Gauss–Legendre two-point quadrature formula and the Gauss–Chebyshev two-point quadrature formulas of the first and second kind.

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On approximately monotone and approximately Hölder functions

Cited by 6 publications

References 25 publications

Characterization of approximately monotone and approximately Hölder functions

Characterization of approximately monotone and approximately Hölder functions

Generalizing the concept of bounded variation

Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

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