Dynamic hardy-copson type inequalities on arbitrary time scales

Time scale calculus addresses the modeling of discrete and continuous time phenomena under a unified framework. This approach provides theoretical unity by eliminating the need to analyze systems separately for continuous and discrete cases. Thus, time scale calculus offers a new perspective in understanding not only the current state but also the past states of processes. Hardy-Copson type inequalities are among the cornerstones of classical analysis and play a critical role in defining the bounds and behavioral constraints between the integral and the derivative of a function. In essence, these inequalities demonstrate the effect of a function's local behavior (derivative/derivative-like expressions) on its global behavior (integral/integral means). Owing to these properties, Hardy-Copson type inequalities continue to be a constant source of inspiration for new mathematical theories. This study will systematically present Hardy-Copson inequalities throughout their historical context, examine them in detail, identify their differences and similarities, and compile a review on the subject. First, Hardy-Copson inequalities obtained in discrete and continuous cases will be presented, followed by dynamic Hardy-Copson inequalities on time scales, which are the general form and combination of discrete and continuous cases. In the first chapter, the fundamental definitions and theorems used in the construction of time scale theory will be introduced. In the following chapter, the historical background of studies on Hardy–Copson type inequalities and the used essential sources will be examined. In the third, fourth, and fifth chapters, which constitute the core of the study, the nabla and delta Hardy–Copson type inequalities valid for m≥0 under the conditions of p>1,01 and 0