Riemann–Liouville Fractional Fundamental Theorem of Calculus and Riemann–Liouville Fractional Polya Integral Inequality and the Generalization to Choquet Integral Case

  • George A. AnastassiouEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 886)


Here we present the right and left Riemann–Liouville fractional fundamental theorems of fractional calculus without any initial conditions for the first time. Then we establish a Riemann–Liouville fractional Polya type integral inequality with the help of generalised right and left Riemann–Liouville fractional derivatives. The amazing fact here is that we do not need any boundary conditions as the classical Polya integral inequality requires. We extend our Polya inequality to Choquet integral setting. See also [2].


  1. 1.
    G.A. Anastassiou, Intelligent Mathematics: Computational Analysis (Springer, Heidelberg, 2011)CrossRefGoogle Scholar
  2. 2.
    G.A. Anastassiou, Riemann-Liouville fractional fundamental theorem of Calculus and Riemann-Liouville Fractional Polya type integral inequality and its extension to Choquet integral setting, Bulletin of Korean Mathematical Society (2019)Google Scholar
  3. 3.
    G. Choquet, Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–295 (1953)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. Denneberg, Non-additive Measure and Integral (Kluwer Academic Publishers, Boston, 1994)CrossRefGoogle Scholar
  5. 5.
    I. Podlubny, Fractional Differentiation Equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  6. 6.
    G. Polya, Ein Mittelwertsatz für Funktionen mehrerer Veränderlichen. Tohoku Math. J. 19, 1–3 (1921)zbMATHGoogle Scholar
  7. 7.
    G. Polya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, vol. I (Springer, Berlin, 1925). (German)CrossRefGoogle Scholar
  8. 8.
    G. Polya, G. Szegö, Problems and Theorems in Analysis, vol. I, Classics in Mathematics (Springer, Berlin, 1972)CrossRefGoogle Scholar
  9. 9.
    G. Polya, G. Szegö, Problems and Theorems in Analysis, vol. I, Chinese edn (1984)Google Scholar
  10. 10.
    F. Qi, Polya type integral inequalities: origin, variants, proofs, refinements, generalizations, equivalences, and applications. RGMIA, Res. Rep. Coll., article no. 20, vol. 16 (2013).
  11. 11.
    M. Sugeno, A note on derivatives of functions with respect to fuzzy measures. Fuzzy Sets Syst. 222, 1–17 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    E.T. Whittaker, G.N. Watson, A Course in Modern Analysis (Cambridge University Press, Cambridge, 1927)zbMATHGoogle Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

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