New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative
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Authors
Bashir Ahmad
- Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Mohamed Jleli
- Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia.
Bessem Samet
- Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia.
Abstract
We say that a function \(f:[a,b]\to \mathbb{R}\) is \((\varphi,\delta)\)-Lipschitzian, where \(\delta\geq 0\) and \(\varphi:[0,\infty)\to [0,\infty)\), if
\[
|f(x)-f(y)|\leq \varphi(|x-y|)+\delta,\quad (x,y)\in [a,b]\times [a,b].
\]
In this work, some Hadamard's type inequalities are established for the class of \((\varphi,\delta)\)-Lipschitzian mappings. Moreover, some
applications to convex functions with a continuous Caputo
fractional derivative are also discussed.
Share and Cite
ISRP Style
Bashir Ahmad, Mohamed Jleli, Bessem Samet, New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 5, 658--671
AMA Style
Ahmad Bashir, Jleli Mohamed, Samet Bessem, New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative. J. Nonlinear Sci. Appl. (2018); 11(5):658--671
Chicago/Turabian Style
Ahmad, Bashir, Jleli, Mohamed, Samet, Bessem. "New integral inequalities and their applications to convex functions with a continuous Caputo fractional derivative." Journal of Nonlinear Sciences and Applications, 11, no. 5 (2018): 658--671
Keywords
- \((\varphi
- \delta)\)-Lipschitzian
- Hadamard's type inequalities
- convex function
- Caputo fractional derivative
- fractional mean value theorem
MSC
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