Simpson and Newton type inequalities for convex functions via newly defined quantum integrals

Abstract

We first establish two new identities, based on the kernel functions with either two section or three sections, involving quantum integrals by using new definition of quantum derivative. Then, some new inequalities related to Simpson's 1/3 formula for convex mappings are provided. In addition, Newton type inequalities, for functions whose quantum derivatives in modulus or their powers are convex, are deduced. We also mention that the results in this work generalize inequalities given in earlier study.

New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions

“…Remark 15 If we set η(κ 2 , κ 1 ) = κ 2 -κ 1 and η(κ 1 , κ 2 ) = κ 1 -κ 2 in (4.31), then the inequality (4.31) reduces to the inequality presented in [12,Remark 5].…”

“…In 2013, Tariboon introduced using classical convexity. Many mathematicians have done studies in q-calculus analysis; the interested reader can check [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Post-quantum Hermite–Hadamard type inequalities for interval-valued convex functions

“…In 2004, Rajkovic gave a definition of the Riemann-type q-integral which was a generalization of Jackson q-integral. In 2013, Tariboon introduced mathematicians have done studies in q-calculus analysis, the interested reader can check [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Some new Simpson's type inequalities for coordinated convex functions in quantum calculus

“…In recent decades, the idea of convex functions has been drastically studied because of its fantastic significance in numerous fields of pure and applied sciences. Theory of inequalities and concept of convex functions are closely related to each other; thus, diverse inequalities can be found within the literature which are proved for convex and differentiable convex functions of single and double variables, see other studies 7‐18 …”

“…Theory of inequalities and concept of convex functions are closely related to each other; thus, diverse inequalities can be found within the literature which are proved for convex and differentiable convex functions of single and double variables, see other studies. [7][8][9][10][11][12][13][14][15][16][17][18] Simpson's rules are well-known techniques for the numerical integration and the numerical estimations of definite integrals. This method is known to be developed by Thomas Simpson (1710-1761).…”