Monotonicity and convexity results for a function through its Caputo fractional derivative-Reference-Cited by-同舟云学术

Monotonicity and convexity results for a function through its Caputo fractional derivative

Published:2017-06-01 Issue:3 Volume:20 Page:818-824
ISSN:1311-0454
Container-title:Fractional Calculus and Applied Analysis
language:en
Short-container-title:

Author:

Al-Refai Mohammed¹

Affiliation:

1. Department of Mathematical Sciences , United Arab Emirates University , P.O.Box. 15551 , Al Ain , UAE

Abstract

Abstract In this paper we discuss certain geometric properties of a function through its Caputo fractional derivative. We show that convexity and monotonicity results can be obtained provided that the fractional derivative of the function is of one sign for some value of α. Analogous result for the global extrema of a function is obtained. However, to release the condition on the Diethelm paper [3] that the fractional derivative of the function is of one sign for all values of α in certain domain, higher order fractional inequalities are required. The applicability of the new results is discussed.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

Link

https://www.degruyter.com/document/doi/10.1515/fca-2017-0042/pdf

Reference6 articles.

1. M. Al-Refai, On the fractional derivatives at extreme points. Electron. J. Qual. Theory Differ. Equ. 2012 (2012), Paper No 55, 1–5.

2. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin, 2010.

3. K. Diethelm, Monotonocity of functions and sign changes of their Caputo derivatives. Fract. Calc. Appl. Anal. 19, No 2 (2016), 561–566; 10.1515/fca-2016-0029; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.

4. J.W. Hanneken. D.M. Vaught, B.N. Narahari, Enumeration of the real zeros of the Mittag-Leffler function Eα(z), 1 < α < 2. In: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Eds.: J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Springer, 2007, 15–26; 10.1007/978-1-4020-6042-7_2.

5. Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351 (2009), 218–223.

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