No abstract

The main of this article are presenting generalized Opial type inequalities which will be defined as theOpial-Jensen inequality for convex function. Further, new Opial type inequalities will be given for functionalsdefined with the help of the Opial inequalities.

The aim of this paper is to establish a generalization and a refinement of Wirtinger's inequality for conformable fractional integrals.

<abstract><p>In this paper, we establish an integral identity involving differentiable functions and generalized fractional integrals. Then, using the newly established identity, we prove some new general versions of Bullen and trapezoidal type inequalities for differentiable convex functions. The main benefit of the newly established inequalities is that they can be converted into similar inequalities for classical integrals, Riemann-Liouville fractional integrals, $ k $-Riemann-Liouville fractional integrals, Hadamard fractional integrals, etc. Moreover, the inequalities presented in the paper are extensions of several existing inequalities in the literature.</p></abstract>