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\( \alpha \)-fractionally convex functions. (English) Zbl 1450.26003

This paper introduces the concept of alpha-convexity in analogy to the differential characterization of convexity but using fractional derivatives. After reading this article, the utility of this concept is not clear to the reviewer as no interesting property was established. Moreover, most of the statements are formally wrong, since the authors denote a function \(f\) by \(f(x)\). This leads to incorrect claims, e.g., \(f(x) > f(a)\) does not imply that \(f\) is a monotonically increasing function when \(a\) is a fixed number.
Further, Theorem 3.6 applies the Leibniz rule to an integrated function that is not continuous in the integration limits. Theorem 3.11–3.12 are based on Theorem 3.10 that is taken from a very odd result in [J. D. Munkhammar, “Fractional calculus and the Taylor-Riemann series”, Undergrad. Math. J. 6, No. 1, Article 6, 19 p. (2005)]. The intereseted reader must take a careful reading of [loc. cit.]. In Theorem 3.13, it is not clear what the contradiction is.

MSC:

26A33 Fractional derivatives and integrals
26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations
52A41 Convex functions and convex programs in convex geometry
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