Fractional calculus is extremely important and should not be undervalued due to its critical role in the theory of inequalities. In this article, different generalized Hermite-Hadamard type inequalities for functions ...
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Fractional calculus is extremely important and should not be undervalued due to its critical role in the theory of inequalities. In this article, different generalized Hermite-Hadamard type inequalities for functions whose modulus of first derivatives are (alpha,s)-convex are presented, via Caputo-Fabrizio integrals. Graphical justifications of main results are presented. Graphs enable us to support our conclusions and show the reliability of our findings. Additionally, some applications to probability theory and numerical integration are also established. As special cases, certain established outcomes from different articles are recaptured. This study acts as a stimulant for future studies, inspiring researchers to investigate more thorough results by utilizing generalized fractional operators and broadening the idea of convexity.
Murota (1995) introduced an M-convex function as a quantitative generalization of the set of integral vectors in an integral base polyhedron as well as an extension of valuated matroid over base polyhedron. Just as a ...
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Murota (1995) introduced an M-convex function as a quantitative generalization of the set of integral vectors in an integral base polyhedron as well as an extension of valuated matroid over base polyhedron. Just as a base polyhedron can be transformed through a network, an M-convex function can be induced through a network. This paper gives a constructive proof for the induction of an M-convex function. The proof is based on the correctness of a simple algorithm, which finds an exchangeable element. We also analyze a behavior of induced functions when they take the value -infinity. (C) 1998 Elsevier Science B.V. All rights reserved.
The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameteriz...
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The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers.
In this paper we generalize a result of Mercer [***. Mercer, A variant of Jensen's inequality, J. Inequal. Pure & Appl. Math. 4 (4) (2003) Article 73. Online: http://***. ***/***?sid=314] on convex functions. ...
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In this paper we generalize a result of Mercer [***. Mercer, A variant of Jensen's inequality, J. Inequal. Pure & Appl. Math. 4 (4) (2003) Article 73. Online: http://***. ***/***?sid=314] on convex functions. A relationship between this result and majorization ordering is pointed out. An extension of Mercer's result to pairs of similarly separable m-tuples is given. In addition, applications are discussed for nonincreasing in mean tuples, star-shaped tuples and convex tuples. (C) 2009 Elsevier Ltd. All rights reserved.
Many converses of Jensen's inequality for convex functions can be found in the literature. Here we give matrix versions, with matrix weights, of these inequalities. Some applications to the Hadamard product of mat...
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Many converses of Jensen's inequality for convex functions can be found in the literature. Here we give matrix versions, with matrix weights, of these inequalities. Some applications to the Hadamard product of matrices are also given. (C) 1997 Academic Press.
In this paper, the authors establish some new estimates for the remainder term of the midpoint, trapezoid, and Simpson formula using functions whose derivatives in absolute value at certain power are s-convex. Some ap...
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In this paper, the authors establish some new estimates for the remainder term of the midpoint, trapezoid, and Simpson formula using functions whose derivatives in absolute value at certain power are s-convex. Some applications to special means of real numbers are provided as well. (C) 2014 Elsevier Inc. All rights reserved.
Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the pr...
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Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we not only recover a significant part of the theory of proximal averages, but also obtain results and properties which have not been previously available.
The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the depe...
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The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results. (C) 2011 Elsevier Ltd. All rights reserved.
In this paper, we derive discrete inequalities of the majorization type for convex functions by using Chebyshev type inequalities for synchronous sequences (cf. Otachel, 2011). Thus, some of the related results from D...
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In this paper, we derive discrete inequalities of the majorization type for convex functions by using Chebyshev type inequalities for synchronous sequences (cf. Otachel, 2011). Thus, some of the related results from Dragomir (2004) and Niezgoda (2008) are unified and extended. Applications for particular convex functions are given. (C) 2014 Elsevier Inc. All rights reserved.
In this paper, first, we prove the weighted Hermite-Hadamard-Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right-sides of weighted Hermite-Hadamard-Merc...
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In this paper, first, we prove the weighted Hermite-Hadamard-Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right-sides of weighted Hermite-Hadamard-Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works.
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