Left and Right Derivatives of Summation Functions Limit and Eulerian Constants Induced by They

Ahmed T Mohammed

doi:10.47134/ppm.v2i3.1747

Authors

Ahmed T Mohammed University of the Alkitab

DOI:

https://doi.org/10.47134/ppm.v2i3.1747

Keywords:

One-sided Derivatives, Limit Summation Functions, Euler-type Constants

Abstract

This study aims to investigate the existence and properties of one-sided derivatives of limit summation functions, particularly in relation to Euler-type constants, within the context of convex and concave real functions. It also seeks to generalize existing theorems related to the differentiability and summability of such functions. The research adopts a theoretical and deductive approach grounded in mathematical analysis. It begins with a comprehensive literature review of foundational concepts such as gamma and zeta functions, convexity, and Euler-Mascheroni constants. Utilizing formal mathematical reasoning, the study develops and proves several new theorems concerning the right and left derivatives of summation functions. The derived results are then validated through a series of examples involving known real functions, including convex and concave functions. The analysis confirms that under specific conditions, one-sided derivatives of summation functions exist and obey certain functional equations. Furthermore, the study demonstrates that sequences related to these derivatives converge under monotonicity assumptions. Applications include generalized inequalities and functional identities related to Euler’s constant, gamma, and zeta functions. Ultimately, this research contributes to the understanding of marginal addition functions and offers new insights into the summability and differentiability of real functions involving Euler-type constants

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