On Some Properties of Three Parameters Mittag-Leffler Functions

Year 2021, , 161 - 169, 31.12.2021

İbrahim Aktaş , Ayşe Topbaş

https://doi.org/10.30931/jetas.1031634

Abstract

In this paper, our main aim is to discuss some monotonic and logarithmic concavity properties of three-parameter Mittag-Leffler function by using its Weierstrassian product representation and some earlier results on power series. In addition, by using the relationships between Mittag-Leffler type functions and some basic functions, we give some specific examples related to the monotonic and logarithmic concavity properties of some trigonometric and hyperbolic functions.

Keywords

Mittag-Leffler function, real zeros of Mittag-Leffler function, monotonicity, logarithmic concavity

References

• [1] Baricz, Á., Prajapati, A., “Radii of starlikeness and convexity of generalized Mittag-Leffler functions”, Mathematical Communication 25(1) (2020) : 117–135.
• [2] Biernacki, M., Krzyż, J., “On the monotonity of certain functionals in the theory of analytic functions”, Annales Universitatis Mariae Curie-Sklodowska, sectio A 9 (1955) : 135–147.
• [3] Gorenflo, R., Kilbas, A. A., Mainardi, F., Rogosin, S. V., “Mittag-Leffler functions, related topics and applications”, Springer, 2014.
• [4] Gupta, I. S., Debnath, L., “Some properties of the Mittag-Leffler functions”, Integral Transforms and Special Functions 18(5) (2007) : 329-336.
• [5] Kumar, H., Pathan, M. A., “On the distribution of non-zero zeros of generalized Mittag-Leffler functions”, International Journal of Engineering Research and Application 6(10) (2016) : 66-71.
• [6] Borzadaran, G. M., Borzadaran, H. M., “Log-concavity property for some well-known distributions”, Surveys in Mathematics and its Applications 6 (2011) : 203-19.
• [7] Peresyolkova, I. N., “On distribution of zeros of generalized functions of Mittag-Leffler type”, Matematychni Studii 13(2) (2000) : 157-64.
• [8] Prabhakar, T. R., “A singular integral equation with a generalized Mittag Leffler function in the kernel”, Yokohama Mathematical Journal 19 (1971) : 7–15.
• [9] Wiman, A., “Über die Nullstellen der Funktionen 𝐸𝐸𝑎𝑎(𝑥𝑥)”, Acta Mathematica 29(1) (1905) : 217-34.