Hermite–Hadamard-type inequalities via n-polynomial exponential-type convexity and their applications
Saad Ihsan Butt, Artion Kashuri, Muhammad Tariq, Jamshed Nasir, Adnan Aslam, Wei Gao
Abstract
Abstract In this paper, we give and study the concept of n -polynomial $(s,m)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:math> -exponential-type convex functions and some of their algebraic properties. We prove new generalization of Hermite–Hadamard-type inequality for the n -polynomial $(s,m)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:math> -exponential-type convex function ψ . We also obtain some refinements of the Hermite–Hadamard inequality for functions whose first derivatives in absolute value at certain power are n -polynomial $(s,m)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:math> -exponential-type convex. Some applications to special means and new error estimates for the trapezoid formula are given.