On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions

Abstract

For μ∈C1(I)\mu \in {C}^{1}\left(I), μ>0\mu \gt 0, and λ∈C(I)\lambda \in C\left(I), where II is an open interval of R{\mathbb{R}}, we consider the set of functions f∈C2(I)f\in {C}^{2}\left(I) satisfying the second-order differential inequality ddtμdfdt+λf≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in II. The considered set includes several classes of generalized convex functions from the literature. In particular, if μ≡1\mu \equiv 1 and λ=k2\lambda ={k}^{2}, k>0k\gt 0, we obtain the class of trigonometrically kk-convex functions, while if μ≡1\mu \equiv 1 and λ=−k2\lambda =-{k}^{2}, k>0k\gt 0, we obtain the class of hyperbolic kk-convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic kk-convex functions.

Keywords

• generalized convexity
• differential inequalities
• trigonometrically k-convex functions
• hyperbolic k-convex functions
• fejér inequality
• characterization
• 26a51
• 26d15
• 34b09