Generalized inequalities for integral operators via several kinds of convex functions

Yue Wang; Ghulam Farid; Babar Khan Bangash; Weiwei Wang; Yue Wang; Ghulam Farid; Babar Khan Bangash; Weiwei Wang

doi:10.3934/math.2020297

AIMS Mathematics

2020, Volume 5, Issue 5: 4624-4643. doi: 10.3934/math.2020297

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Research article

Generalized inequalities for integral operators via several kinds of convex functions

1.
Division of Public Teaching, Jinan Vocational Collage, Jinan, 250103, China
2.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
3.
Department of Mathematics, COMSATS University Islamabad, Attock Campus, Pakistan
4.
The Department of Foundation courses, Shandong Polytechnic, Jinan, 250104, China

Received: 17 March 2020 Accepted: 09 May 2020 Published: 25 May 2020
MSC : 26A51, 26A33, 26D15

This paper investigates the bounds of an integral operator for several kinds of convex functions. By applying definition of (h - m)-convex function upper bounds of left sided (1.12) and right sided (1.13) integral operators are formulated which particularly provide upper bounds of various known conformable and fractional integrals. Further a modulus inequality is investigated for differentiable functions whose derivative in absolute value are (h - m)-convex. Moreover a generalized Hadamard inequality for (h - m)-convex functions is proved by utilizing these operators. Also all the results are obtained for (α, m)-convex functions. Finally some applications of proved results are discussed.
- (h - m)-convex function,
- (α, m)-convex function,
- integral operators,
- fractional integral operators,
- conformable integral operators
Citation: Yue Wang, Ghulam Farid, Babar Khan Bangash, Weiwei Wang. Generalized inequalities for integral operators via several kinds of convex functions[J]. AIMS Mathematics, 2020, 5(5): 4624-4643. doi: 10.3934/math.2020297

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Abstract

This paper investigates the bounds of an integral operator for several kinds of convex functions. By applying definition of (h - m)-convex function upper bounds of left sided (1.12) and right sided (1.13) integral operators are formulated which particularly provide upper bounds of various known conformable and fractional integrals. Further a modulus inequality is investigated for differentiable functions whose derivative in absolute value are (h - m)-convex. Moreover a generalized Hadamard inequality for (h - m)-convex functions is proved by utilizing these operators. Also all the results are obtained for (α, m)-convex functions. Finally some applications of proved results are discussed.

References

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