On new generalized unified bounds via generalized exponentially harmonically s-convex functions on fractal sets

Abstract

Abstract The visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically ( G E H $GEH$ ) convex and G E H $GEH$ s-convex functions on a real linear fractal sets R α $\mathbb{R}^{\alpha }$ ( 0 < α ≤ 1 $0<\alpha \leq 1$ ). Based on these novel ideas, we derive the generalized Hermite–Hadamard inequality, generalized Fejér–Hermite–Hadamard type inequality and Pachpatte type inequalities for G E H $GEH$ s-convex functions. Taking into account the local fractal identity; we establish a certain generalized Hermite–Hadamard type inequalities for local differentiable G E H $GEH$ s-convex functions. Meanwhile, another auxiliary result is employed to obtain the generalized Ostrowski type inequalities for the proposed techniques. Several special cases of the proposed concept are presented in the light of generalized exponentially harmonically convex, generalized harmonically convex and generalized harmonically s-convex. Meanwhile, an illustrative example and some novel applications in generalized special means are obtained to ensure the correctness of the present results. This novel strategy captures several existing results in the corresponding literature. Finally, we suppose that the consequences of this paper can stimulate those who are interested in fractal analysis.

Keywords

• Harmonically convex function
• Exponentially convex function
• Exponentially harmonically s-convex function
• Hermite–Hadamard-Fejér type inequality
• Pachpatte type inequality
• Ostrowski type inequality