New Quantum Estimates of Trapezium-Type Inequalities for Generalized ϕ-Convex Functions

In this paper, a quantum trapezium-type inequality using a new class of function, the so-called generalized ϕ -convex function, is presented. A new quantum trapezium-type inequality for the product of two generalized ϕ -convex functions is provided. The authors also prove an identity for twice q - differentiable functions using Raina’s function. Utilizing the identity established, certain quantum estimated inequalities for the above class are developed. Various special cases have been studied. A brief conclusion is also given.

[1]  C. Bennett,et al.  Interpolation of operators , 1987 .

[2]  Adesanmi Alao Mogbademu,et al.  Some New Hermite-Hadamard Integral Inequalities for Convex functions , 2014 .

[3]  Muhammad Aslam Noor,et al.  Quantum Analogues of Hermite–Hadamard Type Inequalities for Generalized Convexity , 2015 .

[4]  Sorin Olaru,et al.  Convex Lifting: Theory and Control Applications , 2018, IEEE Transactions on Automatic Control.

[5]  Jessada Tariboon,et al.  Quantum integral inequalities on finite intervals , 2014 .

[6]  Wenjun Liu,et al.  Some Quantum Estimates of Hermite-Hadamard Inequalities for Quasi-Convex Functions , 2019, Mathematics.

[7]  Thomas Ernst,et al.  A Comprehensive Treatment of q-Calculus , 2012 .

[8]  Mehmet Zeki Sarikaya,et al.  q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions , 2016 .

[9]  Muhammad Aslam Noor,et al.  Some quantum estimates for Hermite-Hadamard inequalities , 2015, Appl. Math. Comput..

[10]  R. Raina.  ON GENERALIZED WRIGHT'S HYPERGEOMETRIC FUNCTIONS AND FRACTIONAL CALCULUS OPERATORS , 2005 .

[11]  J. Tariboon.  Quantum Calculus , 2020, The Journal of King Mongkut's University of Technology North Bangkok.

[12]  Wenjun Liu,et al.  Some quantum estimates of Hermite-Hadamard inequalities for convex functions , 2017 .

[13]  J. Hadamard,et al.  Etude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann , 1893 .

[14]  Erhan Set,et al.  Generalized Hermite-Hadamard type inequalities involving fractional integral operators , 2017, Journal of Inequalities and Applications.

[15]  H. Gauchman,et al.  Integral inequalities in q-calculus , 2004 .

[16]  J. Jensen.  Sur les fonctions convexes et les inégalités entre les valeurs moyennes , 1906 .

[17]  Artion Kashuri,et al.  Some new hermite-hadamard type inequalities and their applications , 2019, Studia Scientiarum Mathematicarum Hungarica.

[18]  M. Sen,et al.  Some generalizations of Hermite-Hadamard type inequalities. , 2016 .