General Raina fractional integral inequalities on coordinates of convex functions
Dumitru Bǎleanu, Artion Kashuri, Pshtiwan Othman Mohammed, Badreddine Meftah
Abstract
Abstract Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an $(l_{1},h_{1})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> - $(l_{2},h_{2})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> -convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the $(l_{1},h_{1})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> - $(l_{2},h_{2})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> -convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.