On Generalized Strongly Convex Functions and Unified Integral Operators

In this paper, we define a strongly exponentially α , h − m -convex function that generates several kinds of strongly convex and convex functions. The left and right unified integral operators of these functions satisfy some integral inequalities which are directly related to many unified and fractional integral inequalities. From the results of this paper, one can obtain various fractional integral operator inequalities that already exist in the literature.

[1]  Shin Min Kang,et al.  Inequalities for a Unified Integral Operator and Associated Results in Fractional Calculus , 2019, IEEE Access.

[2]  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.

[3]  Daniel Alexandru Ion Some estimates on the Hermite-Hadamard inequality through quasi-convex functions , 2007 .

[4]  Miguel Vivas-Cortez,et al.  Hermite-Hadamard-Fejer Type Inequalities for Strongly (s,m)-Convex Functions with Modulus c, in Second Sense , 2016 .

[5]  G. Farid,et al.  Derivation of bounds of several kinds of operators via (s,m)$(s,m)$-convexity , 2020 .

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

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

[8]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[9]  G. Anastassiou GENERALISED FRACTIONAL HERMITE-HADAMARD INEQUALITIES INVOLVING m-CONVEXITY AND (s,m)-CONVEXITY , 2013 .

[10]  A. Akdemir,et al.  ON h m CONVEXITY AND HADAMARD TYPE INEQUALITIES , 2011, 1103.6163.

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

[12]  G. Jameson,et al.  Refining Jensen's inequality. , 2004 .

[13]  S. Mehmood,et al.  Generalized k-fractional integral inequalities associated with (α,m)$(\alpha ,m)$-convex functions , 2019, Journal of Inequalities and Applications.

[14]  G. Farid Bounds of Riemann-Liouville fractional integral operators , 2020 .

[15]  P. Agarwal Some inequalities involving Hadamard‐type k‐fractional integral operators , 2017 .

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

[17]  S. Mehmood,et al.  Bounds of Riemann-Liouville Fractional Integrals in General Form via Convex Functions and Their Applications , 2018, Mathematics.

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

[19]  S. Dragomir Lebesgue Integral Inequalities of Jensen Type for λ-Convex Functions , 2016, Armenian Journal of Mathematics.

[20]  J. Pečarić,et al.  Refinements of some integral inequalities for unified integral operators , 2021 .

[21]  N. Latif,et al.  Estimations of fractional integral operators for convex functions and related results , 2020 .

[22]  G. Farid A unified integral operator and further its consequences , 2020 .

[23]  Inequalities of Jensen's type for generalized k-g-fractional integrals of function $f$ for which the composite f ○ g^-1 is convex , 2018 .

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

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

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

[27]  J. Pečarić,et al.  Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity , 2020 .

[28]  Q. Ain,et al.  K-fractional integral inequalities of Hadamard type for (h − m)−convex functions , 2020 .

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

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

[31]  Generalized Fractional Integral Operators Involving Mittag-Leffler Function , 2018, Abstract and Applied Analysis.

[33]  Study of fractional integral inequalities involving Mittag-Leffler functions via convexity , 2020 .

[34]  J. Biazar,et al.  Finite integration method with RBFs for solving time-fractional convection-diffusion equation with variable coefficients , 2019 .

[35]  E. Godunova Inequalities for functions of a broad class that contains convex, monotone and some other forms of functions , 1985 .

[36]  Junesang Choi,et al.  Hermite-Hadamard type inequalities for generalized convex functions on fractal sets style , 2018 .

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

[38]  M. Jleli,et al.  Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals , 2017, Journal of inequalities and applications.

[39]  Sanja Varošanec,et al.  On h-convexity , 2007 .

[40]  M. Noor,et al.  Hermite-Hadamard Inequalities for Exponentially Convex Functions , 2018 .

[41]  G. Farid,et al.  Inequalities for a Unified Integral Operator via α,m-Convex Functions , 2020 .

[42]  M. Anwar,et al.  Hermite–Hadamard type inequalities for exponentially p-convex functions and exponentially s-convex functions in the second sense with applications , 2019, Journal of Inequalities and Applications.