On the Fractional Mean Value

TL;DR

This paper explores fractional calculus by extending the classical mean value theorem, introducing fractional critical points, and examining their properties and implications in convexity and time dilation effects.

Contribution

It introduces the concept of fractional critical points and studies their existence, connecting fractional derivatives with convexity and monotonicity in a novel way.

Findings

Sufficient conditions for fractional critical points are established.

An illustrative example related to time dilation is provided.

Connections between convexity and fractional derivatives are analyzed.

Abstract

This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some sufficient conditions for the existence of a critical point is studied and an illustrative example rele- vant to the concept of the time dilation effect is given. The present paper also includes, some connections between convexity (and monotonicity) with fractional derivative in the Riemann-Liouville sense.