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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, coef-
ficient bounds and coefficient estimates
2020 Mathematical Subject Classification: 33E20, 30C45,
30C55
Results & Lemmas (9)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.
Lemma 1. [11] A function 𝑓P 𝒜belongs to the class 𝒢p𝜆,𝛾q if 8 ÿ 𝑛“2 p𝑛` 𝜆𝑛p𝑛´ 1q ´ 𝛾q |𝑎𝑛| ⩽1 ´ 𝛾.
Lemma 1. [11] A function 𝑓P 𝒜belongs to the class 𝒢p𝜆,𝛾q if 8 ÿ 𝑛“2 p𝑛` 𝜆𝑛p𝑛´ 1q ´ 𝛾q |𝑎𝑛| ⩽1 ´ 𝛾.
Lemma 2.
Lemma 2. [8] A function 𝑓P 𝒜belongs to the class 𝒦p𝜆,𝛾q if 8 ÿ 𝑛“2 𝑛p𝑛` 𝜆𝑛p𝑛´ 1q ´ 𝛾q |𝑎𝑛| ⩽1 ´ 𝛾. Starlikeness, convexity,…
Lemma 2. [8] A function 𝑓P 𝒜belongs to the class 𝒦p𝜆,𝛾q if 8 ÿ 𝑛“2 𝑛p𝑛` 𝜆𝑛p𝑛´ 1q ´ 𝛾q |𝑎𝑛| ⩽1 ´ 𝛾. Starlikeness, convexity, close-to-convexity, and some other geomet- ric properties of special functions, such as Bessel, Struve, Wright, Dini, Mittag-Leffler, Miller-Ross, hypergeometric, etc., have been studied by many mathematicians recently (see, for example, [8], [4], [9], [10], [1], [6], [2]). Motivated by the these works, we obtained sufficient conditions for the Rabotnov function to be in th
Lemma 3.
Lemma 3. [9] If 𝑛P N and 𝛼⩾0, then p1 ` 𝛼q𝑛´1p𝑛´ 1q!Γp1 ` 𝛼q ⩽Γ pp1 ` 𝛼q𝑛q.
Lemma 3. [9] If 𝑛P N and 𝛼⩾0, then p1 ` 𝛼q𝑛´1p𝑛´ 1q!Γp1 ` 𝛼q ⩽Γ pp1 ` 𝛼q𝑛q .
Theorem 1.
Theorem 1. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝜆𝛽2 p1 ` 𝛼q2 ` p1 ` 2𝜆q𝛽 1 ` 𝛼 ` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then…
Theorem 1. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝜆𝛽2 p1 ` 𝛼q2 ` p1 ` 2𝜆q𝛽 1 ` 𝛼 ` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then the normalized Rabotnov function R𝛼,𝛽p𝑧q given by (1) belongs to the class 𝒢p𝜆,𝛾q.
Theorem 2.
Theorem 2. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝜆𝛽3 p1 ` 𝛼q3 ` p1 ` 5𝜆q𝛽2 p1 ` 𝛼q2 ` p3 ` 4𝜆´ 𝛾q𝛽 1 ` 𝛼 ` 1…
Theorem 2. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝜆𝛽3 p1 ` 𝛼q3 ` p1 ` 5𝜆q𝛽2 p1 ` 𝛼q2 ` p3 ` 4𝜆´ 𝛾q𝛽 1 ` 𝛼 ` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then the normalized Rabotnov function R𝛼,𝛽p𝑧q given by (1) belongs to
Corollary 1.
Corollary 1. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝛽 1 ` 𝛼` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then the normalized Rabotnov…
Corollary 1. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝛽 1 ` 𝛼` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then the normalized Rabotnov function R𝛼,𝛽p𝑧q given by (1) belongs to the class 𝒮˚ p𝛾q.
Corollary 2.
Corollary 2. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝛽2 p1 ` 𝛼q2 ` p3 ´ 𝛾q𝛽 1 ` 𝛼 ` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then…
Corollary 2. Let 𝛼⩾0 and 𝛽ą 0. If the following condition is satisfied: 𝑒 𝛽 1`𝛼 ˆ 𝛽2 p1 ` 𝛼q2 ` p3 ´ 𝛾q𝛽 1 ` 𝛼 ` 1 ´ 𝛾 ˙ ⩽2p1 ´ 𝛾q, then the normalized Rabotnov function R𝛼,𝛽p𝑧q given by (1) belongs to the class 𝒞p𝛾q. If we take 𝜆“ 0 and 𝛾“ 0 in our theorems, the results of the calcu- lations for obtained inequalities coincide with the theorems given by Eker
Corollary 3.
Corollary 3. Let 𝛼⩾0 and 𝛽ą 0. If 𝛼ą 𝛽 𝑊p2𝑒q´1 ´ 1, where 𝑊is the Lambert 𝑊function, then the normalized Rabotnov function R𝛼,𝛽p𝑧q is…
Corollary 3. Let 𝛼⩾0 and 𝛽ą 0. If 𝛼ą 𝛽 𝑊p2𝑒q´1 ´ 1, where 𝑊is the Lambert 𝑊function, then the normalized Rabotnov function R𝛼,𝛽p𝑧q is starlike in U.
Corollary 4.
Corollary 4. Let 𝛼⩾0 and 𝛽ą 0. If 𝛽 1`𝛼ă 0.199496, then the normal- ized Rabotnov function R𝛼,𝛽p𝑧q is convex in U. References [1] Bansal…
Corollary 4. Let 𝛼⩾0 and 𝛽ą 0. If 𝛽 1`𝛼ă 0.199496, then the normal- ized Rabotnov function R𝛼,𝛽p𝑧q is convex in U. References [1] Bansal D., Prajapat J. K. Certain geometric properties of the Mittag-Leffler functions. Complex Var. Elliptic Equ., 2016, vol. 61(3), pp 338–350. DOI: https://doi.org/10.1080/17476933.2015.1079628 [2] Baricz A. Geometric properties of generalized Bessel functions. Publ Math Debrecen., 2008, vol. 73, pp 155–178. [3] Duren P. L. Univalent Functions. Grundlehren der Math
Function classes studied:
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