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Ufimsk. Mat. Zh., 2021, Volume 13, Issue 1, Pages 119–130 (Mi ufa548)  

Generalization of Hadamard-type trapezoid inequalities for fractional integral operators

B. Bayraktar, M. Emin Özdemir

Bursa Uludag University, Faculty of Education, Gorukle Campus, 16059, Bursa, Turkey

Abstract: The role of convexity theory in applied problems, especially in optimization problems, is well known. The integral Hermite-Hadamard inequality has a special place in this theory since it provides an upper bound for the mean value of a function. In solving applied problems from different fields of science and technology, along with the classical integro-differential calculus, fractional calculus plays an important role. A lot of research is devoted to obtaining an upper bound in the Hermite-Hadamard inequality using operators of fractional calculus.
The article formulates and proves the identity with the participation of the fractional integration operator. Based on this identity, new generalized Hadamard-type integral inequalities are obtained for functions for which the second derivatives are convex and take values at intermediate points of the integration interval. These results are obtained using the convexity property of a function and two classical integral inequalities, the Hermite-Hadamard integral inequality and its other form, the power mean inequality. It is shown that the upper limit of the absolute error of inequality decreases in approximately $n^{2}$ times, where $n$ is the number of intermediate points. In a particular case, the obtained estimates are consistent with known estimates in the literature. The results obtained in the article can be used in further researches in the integro-differential fractional calculus.

Keywords: convexity, Hermite–Hadamard inequality, Hölder inequality, power–mean inequality, Riemann–Liouville fractional Integrals.

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English version:
Ufa Mathematical Journal, 2021, 13:1, 119–130 (PDF, 354 kB); https://doi.org/10.13108/2021-13-1-119

Bibliographic databases:

UDC: 517.518.86, 517.218.244, 517.927.2
MSC: 26A51, 26D15
Received: 01.04.2020
Language:

Citation: B. Bayraktar, M. Emin Özdemir, “Generalization of Hadamard-type trapezoid inequalities for fractional integral operators”, Ufimsk. Mat. Zh., 13:1 (2021), 119–130; Ufa Math. J., 13:1 (2021), 119–130

Citation in format AMSBIB
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\pages 119--130
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\pages 119--130
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