Exponential trigonometric convex functions and Hermite-Hadamard type inequalities
In this paper, the authors introduce a new class of exponential trigonometric convex functions and define it as follows: A function \(f : I \subset \mathbb{R} \to \mathbb{R}\) is called an exponential trigonometric convex function if for every \(a,b \in I, t\in [0,1],\) \[ f(ta + (1-t)b)\le \frac{\sin\frac{\pi t}{2}}{e^{1-t}}f(a)+\frac{\cos\frac{\pi t}{2}}{e^t}f(b).\] The algebraic properties of this new class of functions are studied and well discussed. Using this newly introduced class of functions, the authors derive and prove the Hermite-Hadamard-type inequalities for this class of functions. Furthermore, some refinements of the Hermite-Hadamard-type inequalities for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, are shown to be exponential trigonometric convex function.
exponential trigonometric convex functions
scientific article; zbMATH DE number 7438324