Inequalities for a Unified Integral Operator and Associated Results in Fractional Calculus

Integral operators are useful in real analysis, mathematical analysis, functional analysis and other subjects of mathematical approach. The goal of this paper is to study a unified integral operator via convexity. By using convexity and conditions of unified integral operators, bounds of these operators are obtained. Furthermore consequences of these results are discussed for fractional and conformable integral operators.

[1]  Dumitru Baleanu,et al.  On a new class of fractional operators , 2017, Advances in Difference Equations.

[2]  E. Beckenbach.  CONVEX FUNCTIONS , 2007 .

[3]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[4]  H. Srivastava,et al.  THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS. NORTH-HOLLAND MATHEMATICS STUDIES , 2006 .

[5]  Muhammad Adil Khan,et al.  Generalized conformable fractional operators , 2019, J. Comput. Appl. Math..

[6]  AHMAD W. FARAJ,et al.  A GENERALIZATION OF MITTAG-LEFFLER FUNCTION AND INTEGRAL OPERATOR ASSOCIATED WITH FRACTIONAL CALCULUS , 2012 .

[7]  Hari M. Srivastava,et al.  Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , 2009, Appl. Math. Comput..

[8]  Ghulam Farid.  Some Riemann–Liouville fractional integral inequalities for convex functions , 2018 .

[9]  W. Marsden.  I and J , 2012 .

[10]  Josip Pečarić,et al.  A further extension of Mittag-Leffler function , 2018, Fractional Calculus and Applied Analysis.

[11]  Ghulam Farid.  Existence of an integral operator and its consequences in fractional and conformable integrals , 2019, Open Journal of Mathematical Sciences.

[12]  Dumitru Baleanu,et al.  On fractional calculus with general analytic kernels , 2019, Appl. Math. Comput..

[13]  Shin Min Kang,et al.  Generalized Riemann-Liouville $k$ -Fractional Integrals Associated With Ostrowski Type Inequalities and Error Bounds of Hadamard Inequalities , 2018, IEEE Access.

[14]  Zoubir Dahmani,et al.  (k, s)-Riemann-Liouville fractional integral and applications , 2016 .

[15]  D. Varberg.  Convex Functions , 1973 .

[16]  Udita N. Katugampola,et al.  Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals ☆ , 2016, 1609.04774.

[17]  Maysaa Mohamed Al Qurashi,et al.  The extended Mittag-Leffler function via fractional calculus , 2017 .

[18]  MUHAMMAD NAWAZ NAEEM.  CHEBYSHEV TYPE INTEGRAL INEQUALITIES FOR GENERALIZED k-FRACTIONAL CONFORMABLE INTEGRALS , 2018 .

[19]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[20]  Asif Waheed,et al.  Generalized fractional inequalities for quasi-convex functions , 2019, Advances in Difference Equations.

[21]  Shahid Mubeen,et al.  k -Fractional Integrals and Application , 2012 .