Trapezoidal ( p , q )-Integral Inequalities Related to ( η 1 , η 2 )-convex Functions with Applications

In this paper, the authors introduce the ( p , q )-trapezoidal integral inequalities, which are the ( p , q )-analogues of the recently introduced q -trapezoidal integral inequalities. We derive a new ( p , q )-integral identity for twice ( p , q )-differentiable function. Utilizing this as an auxiliary result, we establish several new ( p , q )-trapezoidal type integral inequalities for the function whose absolute value of twice ( p , q )-derivative is ( η 1 , η 2 )-convex functions. Some special means of real numbers are also given. At the end, we give brief conclusion. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.