Skip to main content
SpringerLink
Log in

Monotonicity and Convexity Results for a Function Through Its Caputo Fractional Derivative

  • Short Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

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 a in certain domain, higher order fractional inequalities are required. The applicability of the new results is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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; DOI: 10.1007/978-1-4020-6042-7_2.

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, Academic Press, San Diego, 1993.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Mohammed Al-Refai

Authors
  1. Mohammed Al-Refai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed Al-Refai.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Al-Refai, M. Monotonicity and Convexity Results for a Function Through Its Caputo Fractional Derivative. FCAA 20, 818–824 (2017). https://doi.org/10.1515/fca-2017-0042

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2017-0042

MSC 2010

Key Words and Phrases

Search

Navigation