Some geometric properties of the generalized $k$-Bessel functions

Büşra Korkmaz; İbrahim Aktaş

doi:10.26637/mjm1301/003

Abstract

In the present work, we investigate certain geometrical properties of a normalized form of $k$-Bessel function in the open disk $\mathbb{B}_\frac{1}{2}=\left\lbrace \rho\in\mathbb{C}:\left|\rho\right|<\frac{1}{2} \right\rbrace $. Also, we determine some sufficient conditions on the parameters such that the mentioned function is uniform convex and parabolic starlike, respectively. In addition, we present some corollaries regarding classical Bessel function and modified Bessel functions of the first kind for some special values of parameters. Finally, we give some examples for our results and support them with graphs.

Keywords:

Analytic function, $k$-Bessel function, Starlikeness, convexity, uniform convexity, parabolic starlikeness.

Mathematics Subject Classification:

30C45, 33E50

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Büşra Korkmaz Department of Mathematics, Kam˙ıl ¨Ozda˘g Science Faculty, Karamano˘glu Mehmetbey University, Yunus Emre Campus, 70100, Karaman, T¨urkiye.
İbrahim Aktaş Department of Mathematics, Kam˙ıl ¨Ozda˘g Science Faculty, Karamano˘glu Mehmetbey University, Yunus Emre Campus, 70100, Karaman, T¨urkiye. https://orcid.org/0000-0003-4570-4485

Pages: 15-25
Date Published: 01-01-2025
Vol. 13 No. 01 (2025): Malaya Journal of Matematik (MJM)

˙I. AKTAS¸ AND L.-I. COT´IRLA, Certain Geometrical Properties and Hardy Space of Generalized k-Bessel Functions, Symmetry, 16 (12) (2024), 1597, https://doi.org/10.3390/sym16121597. DOI: https://doi.org/10.3390/sym16121597

˙I. AKTAS¸, On some geometric properties and Hardy class of q-Bessel functions, AIMS Math., 5 (4) (2020), 3156–3168. DOI: https://doi.org/10.3934/math.2020203

˙I. AKTAS¸, Certain geometric properties of a normalized hyper-Bessel function, Facta Univ. Ser. Math. Inf., 35(1) (2020), 179–86. DOI: https://doi.org/10.22190/FUMI2001179A

˙I. AKTAS¸, ´A. BARICZ AND S. SINGH, Geometric and monotonic properties of hyper-Bessel functions, Ramanujan J., 51(1-2) (2020), 275–295. DOI: https://doi.org/10.1007/s11139-018-0105-9

D. BANSAL AND J.K. PRAJAPAT, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61(3) (2016), 338–350. DOI: https://doi.org/10.1080/17476933.2015.1079628

D. BANSAL, M.K. SONI AND A. SONI, Certain geometric properties of the modified Dini function, Anal. Math. Phys., 9 (2019), 1383–1392. DOI: https://doi.org/10.1007/s13324-018-0243-7

S. DAS AND K. MEHREZ, Geometric properties of the four parameters Wright function, J. Contemp. Math. Anal., 57(1) (2022), 43–58. DOI: https://doi.org/10.3103/S1068362322010058

E. DENIZ AND S¸ . G ¨OREN, Geometric properties of generalized Dini functions, Honam Math. J., 41(1) (2019), 101–116.

R. D´IAZ AND E. PARIGUAN, On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Mathem´aticas, 15 (2) (2007), 179–192.

M.U. DIN, M. RAZA, S. HUSSAIN AND M. DARUS, Certain geometric properties of generalized Dini functions, Journal of Function Spaces, Volume 2018, Article ID 2684023, 9 pages DOI: https://doi.org/10.1155/2018/2684023

A.W. GOODMAN, On uniformly convex functions, J. Math. Anal. Appl., 155 (1991), 364–370. DOI: https://doi.org/10.1016/0022-247X(91)90006-L

A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 57 (1991), 87–92. DOI: https://doi.org/10.4064/ap-56-1-87-92

T.H. MACGREGOR, The radius of univalence of certain analytic functions II, Proc. Am. Math. Soc., 14(3) (1963), 521–524. DOI: https://doi.org/10.1090/S0002-9939-1963-0148892-5

T.H. MACGREGOR, A class of univalent functions, Proc. Am. Math. Soc., 15(2) (1964), 311–317. DOI: https://doi.org/10.1090/S0002-9939-1964-0158985-5

K. MEHREZ, Some geometric properties of a class of functions related to the Fox–Wright functions, Banach J. Math. Anal., 14(3) (2020), 1222-1240. DOI: https://doi.org/10.1007/s43037-020-00059-w

K. MEHREZ, S. DAS AND A. KUMAR, Geometric properties of the products of modified Bessel functions of the first kind, Bull. Malays. Math. Sci. Soc., 6(2021), 1–9.

K. MEHREZ AND D. SOURAV, On some geometric properties of the Le Roy-type Mittag-Leffler function, Hacet. J. Math. Stat., 51(4) (2022), 1085–1103.

S.R. MONDAL AND M.S. AKEL, Differential equation and inequalities of the generalized k-Bessel functions, J. Inequal. Appl., 2018:175. doi: 10.1186/s13660-018-1772-1 DOI: https://doi.org/10.1186/s13660-018-1772-1

S. NOREEN, M. RAZA, M.U. DIN AND S. HUSSAIN, On certain geometric properties of normalized Mittag-Leffler functions, UPB Sci. Bull. Ser. A Appl. Math. Phys., 81 (2019), 167–174.

S. NOREEN, M. RAZA, J.L. LIU AND M. ARIF, Geometric properties of normalized Mittag–Leffler functions, Symmetry, 11(1) (2019), 45. DOI: https://doi.org/10.3390/sym11010045

S. NOREEN, M. RAZA AND S.N. MALIK, Certain geometric properties of Mittag-Leffler functions, J. Inequal. Appl., 2019: 1–5. DOI: https://doi.org/10.1186/s13660-019-2044-4

J.K. PRAJAPAT, S. MAHARANA AND D. BANSAL, Radius of Starlikeness and Hardy Space of Mittag-Leffer Functions, Filomat, 32(18)(2018), 6475–6486. DOI: https://doi.org/10.2298/FIL1818475P

V. RAVICHANDRAN, On uniformly convex functions, Ganita, 53(2) (2002), 117–124.

M. RAZA, D. BREAZ,S. MUSTHAG, L.I. COTˆIRL ˇA, F.M.O. TAWFIQ AND E. RAPEANU,

Geometric properties and Hardy spaces of Rabotnov fractional exponential functions, Fractal and Fractional, 8 (5) (2024), https://doi.org/10.3390/fractalfract8010005. DOI: https://doi.org/10.3390/fractalfract8010005

F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196. DOI: https://doi.org/10.1090/S0002-9939-1993-1128729-7

S. SARKAR, S. DAS AND S. MONDAL, Geometric properties of normalized Galu´e type Struve function, Symmetry, 16 (2024), 211, https://doi.org/10.3390/sym16020211 DOI: https://doi.org/10.3390/sym16020211

E. TOKLU, Radii of starlikeness and convexity of generalized k-Bessel functions, Journal of Applied Analysis, 29(1) (2023), 171–185. DOI: https://doi.org/10.1515/jaa-2022-2005

E. TOKLU, On some geometric results for generalized k-Bessel functions, Demonstratio Mathematica, 56(1) (2023), 20220235. https://doi.org/10.1515/dema-2022-0235. DOI: https://doi.org/10.1515/dema-2022-0235

G. N. WATSON, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.