Bounds, Decay, and Integrals of Non-Increasing Functions: A Comprehensive Analysis.

This scientific article investigates non-increasing functions and their integral properties, shedding light on their behavior. By examining ∫∞ 0 h(s)ds = 1 − l > 0. Furthermore, examining h(t) = ∫ ∞ t E α+1(s)ds about E(t) and T > 0, we establish ∫ ∞ t E α+1(s)ds ≤ T Eα (0)E(t), showing how E(t) influences h(t). For t ≥ T, E α+1(t) follows a power-law decay. Our study unveils a set of inequalities providing insights into these relationships, offering potential applications in control theory, differential equations, and dynamical systems. [ABSTRACT FROM AUTHOR]