Certain properties of a class of analytic functions involving the Mathieu type power series

Abdulaziz Alenazi; Khaled Mehrez; Abdulaziz Alenazi; Khaled Mehrez

doi:10.3934/math.20231584

AIMS Mathematics

2023, Volume 8, Issue 12: 30963-30980. doi: 10.3934/math.20231584

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Certain properties of a class of analytic functions involving the Mathieu type power series

Abdulaziz Alenazi ¹,
Khaled Mehrez ^{2
,
,}

1.
Department of Mathematics, College of Science, Northern Border University, Arar, Saudi Arabia
2.
Department of Mathematics, Kairouan Preparatory Institute for Engineering Studies, University of Kairouan, Kairouan, Tunisia

Received: 01 August 2023 Revised: 31 October 2023 Accepted: 03 November 2023 Published: 17 November 2023
MSC : 33E20, 40A10, 30C45

In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved.
- generalized Mathieu-type series,
- analytic function,
- univalent function,
- starlike function,
- close-to-convex function,
- convex function
Citation: Abdulaziz Alenazi, Khaled Mehrez. Certain properties of a class of analytic functions involving the Mathieu type power series[J]. AIMS Mathematics, 2023, 8(12): 30963-30980. doi: 10.3934/math.20231584

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Abstract

In this paper, we studied some geometric properties of a class of analytic functions related to the generalized Mathieu type power series. Furthermore, we have identified interesting consequences and some examples accompanied by graphical representations to illustrate the results achieved.

References

[1]	I. Aktas, On some geometric properties and Hardy class of $q$-Bessel functions, AIMS Math., 5 (2020), 3156–3168. https://doi:10.3934/math.2020203 doi: 10.3934/math.2020203
[2]	I. Aktas, H. Orhan, Bounds for radii of convexity of some $q$-bessel functions, Bull. Korean Math. Soc., 57 (2020), 355–369. https://doi.org/10.4134/BKMS.b190242 doi: 10.4134/BKMS.b190242
[3]	D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61 (2016), 338–350. https://doi.org/10.1080/17476933.2015.1079628 doi: 10.1080/17476933.2015.1079628
[4]	Á. Baricz, Generalized Bessel Functions of the First Kind, Berlin: Springer, 2010. https://doi.org/10.1007/978-3-642-12230-9
[5]	K. Mehrez, D. Bansal, Geometric properties of normalized Le Roy-type Mittag-Leffler functions, J. Contemp. Math. Anal., 57 (2022), 344–357. https://doi.org/10.3103/S1068362322060061 doi: 10.3103/S1068362322060061
[6]	K. Mehrez, Some geometric properties of a class of functions related to the Fox-Wright functions, Banach J. Math. Anal., 14 (2020), 1222–1240. https://doi.org/10.1007/s43037-020-00059-w doi: 10.1007/s43037-020-00059-w
[7]	K. Mehrez, Study of the analytic function related to the Le-Roy-type Mittag-Leffler function, Ukr. Math. J., 75 (2023), 628–649. https://doi.org/10.1007/s11253-023-02225-3 doi: 10.1007/s11253-023-02225-3
[8]	K. Mehrez, S. Dias, On some geometric properties of the Le Roy-type Mittag-Leffler function, Hacet. J. Math. Stat., 51 (2022), 1085–1103. https://doi.org/10.15672/hujms.989236 doi: 10.15672/hujms.989236
[9]	K. Mehrez, S. Das, A. Kumar, Geometric properties of the products of modified Bessel functions of the first kind, Bull. Malays. Math. Sci. Soc., 44 (2021), 2715–2733. https://doi.org/10.1007/s40840-021-01082-2 doi: 10.1007/s40840-021-01082-2
[10]	S. Noreen, M. Raza, M. U. Din, S. Hussain, On certain geometric properties of normalized Mittag-Leffler functions, UPB Sci. Bull. Ser. A, 81 (2019), 167–174.
[11]	S. Noreen, M. Raza, J. L. Liu, M. Arif, Geometric properties of normalized Mittag-Leffler functions, Symmetry, 11 (2019), 45. https://doi.org/10.3390/sym11010045 doi: 10.3390/sym11010045
[12]	S. Noreen, M. Raza, S. N. Malik, Certain geometric properties of Mittag-Leffler functions, J. Inequal. Appl., 2019 (2019), 94. https://doi.org/10.1186/s13660-019-2044-4 doi: 10.1186/s13660-019-2044-4
[13]	E. L. Mathieu, Traitée de Physique Mathématique, VI-VII: Théory de l'Elasticité des Corps Solides (Part 2), Paris: Gauthier-Villars, 1890.
[14]	K. Schröder, Das Problem der eingespannten rechteckigen elastischen Platte Ⅰ.: Die biharmonische Randwertaufgabe für das Rechteck, Math. Anal., 121 (1949), 247–326.
[15]	P. H. Diananda, Some inequalities related to an inequality of Mathieu, Math. Ann., 250 (1980), 95–98. https://doi.org/10.1007/BF02599788 doi: 10.1007/BF02599788
[16]	S. Gerhold, F. Hubalek, Z. Tomovski, Asymptotics of some generalized Mathieu series, Math. Scand., 126 (2020), 424–450. https://doi.org/10.7146/math.scand.a-121106 doi: 10.7146/math.scand.a-121106
[17]	D. Bansal, J. Sokol, Geometric properties of Mathieu-type power series inside unit disk, J. Math. Inequal., 13 2019,911-918. https://doi.org/10.7153/jmi-2019-13-64 doi: 10.7153/jmi-2019-13-64
[18]	S. Gerhold, Z. Tomovski, D. Bansal, A. Soni, Geometric properties of Some generalized Mathieu Power Series inside the unit disk, Axioms, 11 (2022), 568. https://doi.org/10.3390/axioms11100568 doi: 10.3390/axioms11100568
[19]	P. L. Duren, Univalent Functions, New York: Springer-Verlag, 1983.
[20]	W. Kaplan, Close to convex schlicht functions, Michigan Math. J., 2 (1952), 169–185. https://doi.org/10.1307/mmj/1028988895 doi: 10.1307/mmj/1028988895
[21]	S. Ponnusamy, Univalence of Alexander transform under new mapping properties, Complex Variables Theory Appl., 30 (1996), 55–68. https://doi.org/10.1080/17476939608814911 doi: 10.1080/17476939608814911
[22]	S. Ozaki, On the theory of multivalent functions. Sci. Rep. Tokyo Bunrika Daigaku, 2 (1935), 167–188.
[23]	L. Fejér, Untersuchungen über Potenzreihen mit mehrfach monotoner Koeffizientenfolge, Acta Litt. Sci., 8 (1936), 89–115.
[24]	T. H. MacGregor, The radius of univalence of certain analytic functions Ⅱ, Proc. Amer. Math. Soc., 14 (1963), 521–524. https://doi.org/10.2307/2033833 doi: 10.2307/2033833
[25]	T. H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc., 15 (1964), 311–317. https://doi.org/10.2307/2034061 doi: 10.2307/2034061

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