Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-α integrals and Montgomery identity

Abstract

In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.

Keywords

• gr$ \ddot{u} $ss-type inequality
• diamond-$ \alpha $ integral
• ostrowski inequality
• trapezoid inequality
• montgomery identity