• Title, Summary, Keyword: Riemann integral

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RIEMANN-LIOUVILLE FRACTIONAL FUNDAMENTAL THEOREM OF CALCULUS AND RIEMANN-LIOUVILLE FRACTIONAL POLYA TYPE INTEGRAL INEQUALITY AND ITS EXTENSION TO CHOQUET INTEGRAL SETTING

  • Anastassiou, George A.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1423-1433
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    • 2019
  • Here we present the right and left Riemann-Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann-Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann-Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting.

THE RIEMANN-STIELTJES DIAMOND-ALPHA INTEGRAL ON TIME SCALES

  • Zhao, Dafang;You, Xuexiao;Cheng, Jian
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.1
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    • pp.53-63
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    • 2015
  • In this paper, we define and study the Riemann-Stieltjes diamond-alpha integral on time scales. Many properties of this integral will be obtained. The Riemann-Stieltjes diamond-alpha integral contains the Riemann{Stieltjes integral and diamond-alpha integral as special cases.

ON THE OSTROWSKI INEQUALITY FOR THE RIEMANN-STIELTJES INTEGRAL ${\int}_a^b$ f (t) du (t), WHERE f IS OF HÖLDER TYPE AND u IS OF BOUNDED VARIATION AND APPLICATIONS

  • DRAGOMIR, S.S.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.1
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    • pp.35-45
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    • 2001
  • In this paper we point out an Ostrowski type inequality for the Riemann-Stieltjes integral ${\int}_a^b$ f (t) du (t), where f is of p-H-$H{\ddot{o}}lder$ type on [a,b], and u is of bounded variation on [a,b]. Applications for the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also given.

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THE DENJOY EXTENSION OF THE RIEMANN INTEGRAL

  • Park, Jae Myung;Kim, Soo Jin
    • Journal of the Chungcheong Mathematical Society
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    • v.9 no.1
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    • pp.101-106
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    • 1996
  • In this paper, we will consider the Denjoy-Riemann integral of functions mapping a closed interval into a Banach space. We will show that a Riemann integrable function on [a, b] is Denjoy-Riemann integrable on [a, b] and that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b].

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Development of the Integral Concept (from Riemann to Lebesgue) (적분개념의 발달 (리만적분에서 르베그적분으로의 이행을 중심으로))

  • Kim, Kyung-Hwa
    • Journal for History of Mathematics
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    • v.21 no.3
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    • pp.67-96
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    • 2008
  • In the 19th century Fourier and Dirichlet studied the expansion of "arbitrary" functions into the trigonometric series and this led to the development of the Riemann's definition of the integral. Riemann's integral was considered to be of the highest generality and was discussed intensively. As a result, some weak points were found but, at least at the beginning, these were not considered as the criticism of the Riemann's integral. But after Jordan introduced the theory of content and measure-theoretic approach to the concept of the integral, and after Borel developed the Jordan's theory of content to a theory of measure, Lebesgue joined these two concepts together and obtained a new theory of integral, now known as the "Lebesgue integral".

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NOTE ON CAHEN′S INTEGRAL FORMULAS

  • Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.17 no.1
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    • pp.15-20
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    • 2002
  • We present an explicit form for a class of definite integrals whose special cases include some definite integrals evaluated, over a century ago, by Cahen who made use of an appropriate contour integral for the integrand of a well-known integral representation of the Riemann Zeta function given in (3). Furthermore another analogous class of definite integral formulas and some identities involving Riemann Zeta function and Euler numbers En are also obtained as by-products.

COMPARISON THEOREMS IN RIEMANN-FINSLER GEOMETRY WITH LINE RADIAL INTEGRAL CURVATURE BOUNDS AND RELATED RESULTS

  • Wu, Bing-Ye
    • Journal of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.421-437
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    • 2019
  • We establish some Hessian comparison theorems and volume comparison theorems for Riemann-Finsler manifolds under various line radial integral curvature bounds. As their applications, we obtain some results on first eigenvalue, Gromov pre-compactness and generalized Myers theorem for Riemann-Finsler manifolds under suitable line radial integral curvature bounds. Our results are new even in the Riemannian case.

ON A SEQUENCE OF KANTOROVICH TYPE OPERATORS VIA RIEMANN TYPE q-INTEGRAL

  • Bascanbaz-Tunca, Gulen;Erencin, Aysegul;Tasdelen, Fatma
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.303-315
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    • 2014
  • In this work, we construct Kantorovich type generalization of a class of linear positive operators via Riemann type q-integral. We obtain estimations for the rate of convergence by means of modulus of continuity and the elements of Lipschitz class and also investigate weighted approximation properties.