CC BY
41Probl. Anal. Issues Anal. Vol. 12 (30), No 2, 2023, pp. 97-106
DOI: 10.15393/j3.art.2023.12490
97
UDC 517.54
S. SUmer Eker, B. §eker, S. Ece
ON NORMALIZED RABOTNOV FUNCTION ASSOCIATED WITH CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS
Abstract. In this paper, we investigate some sufficient conditions for the normalized Rabotnov function to be in certain subclasses of analytic and univalent functions. The usefulness of the results is depicted by some corollaries and examples.
Key words: Rabotnov function, univalent, starlike, convex, coefficient bounds and coefficient estimates
2020 Mathematical Subject Classification: 33E20, 30C45, 30C55
1. Introduction. The well-known Mittag-Leffler function Ea(z) and its two-parameter version Ea>K(z), which are defined, respectively, by (see, for example, [5]):
8 zn 8 zn
Ea{z) = Y -—T and Ea,K(z )=y ^-■-T, {z,a,K 6 C).
1 (an + 1) + n)
One can see that this series converges in the whole complex plane for all Re(a) > 0. For all Re(a) < 0, this power series diverges everywhere on C\{0}. For Re(a) = 0, the radius of convergence is equal to p = ei 1Im"I. Applications and generalizations of the Mittag-Leffler function have an important place in physics, biology, chemistry, engineering, and other applied sciences.
In 1948, Yu. N. Rabotnov, who worked in solid mechanics, including plasticity, creep theory, hereditary mechanics, failure mechanics, nonelas-tic stability, composites, and shell theory, introduced a special function applied in viscoelasticity [7]. This function, known today as the Rabotnov fractional exponential function, or briefly the Rabotnov function, is
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