Univalence of some integral operators involving Rabotnov fractional exponential function

Basem A. Frasin

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 6, pp. 710-720.

Résumé

In this paper, we introduce three new integral operators involving normalized Rabotnov fractional exponential functions $\mathbb{R}_{\alpha,\beta}(z)$. Furthermore, we shall find sufficient conditions for these integral operators. Finally, some special cases are deduced for different values of $\alpha$ and $\beta$.

Détail

Keywords: analytic functions, integral operators, Rabotnov function.

Bibliographie

[1] A.Amourah, A.Ildawish, K.R.Alhindi, B.A.Frasin, “An application of Rabotnov functions on certain subclasses of bi-univalent functions”, Axioms, 11 (2022), 680 | DOI

[2] I.Aktas, A.Baricz, N.Yagmur, “Bounds for the radii of univalence of some special functions”, Mathematical Inequalities $\$ Applications, 20:3 (2017), 825–843 | DOI | MR | Zbl

[3] M.Arif, M.Raza, “Some properties of an integral operator defined by Bessel functions”, Acta Univ. Apulensis, 26 (2011), 69–74 | MR | Zbl

[4] A.A.Attiya, “Some applications of Mittag-Leffler function in the unit disk”, Filomat, 30:7 (2016), 2075–2081 | DOI | MR | Zbl

[5] D.Bansal, J.K.Prajapat, “Certain geometric properties of the Mittag-Leffler functions”, Complex Var. Elliptic Equ., 61:3 (2016), 338–350 | DOI | MR | Zbl

[6] Á.Baricz, B.A.Frasin, “Univalence of integral operators involving Bessel functions”, Appl. Math. Lett., 23 (2010), 371–276 | DOI | MR

[7] Á.Baricz, M.Caglar, E.Deniz, “Starlikeness of Bessel functions and their derivatives”, Mathematical Inequalities Applications, 19:2 (2016), 439–449 | DOI | MR | Zbl

[8] D.Breaz, N.Breaz, H.M.Srivastava, “An extension of the univalent condition for a family of integral operators”, Appl. Math. Lett., 22 (2009), 41–44 | DOI | MR | Zbl

[9] D.Breaz, H.Ö.Güney, “On the univalence criterion of a general integral operator”, J. Inequal. Appl., 2008, 702715 | DOI | MR | Zbl

[10] S.Bulut, “Univalence preserving integral operators defined by generalized Al-Oboudi differential operators”, An. St. Univ. Ovidius Constata, 17 (2009), 37–50 | MR | Zbl

[11] E.Deniz, “On the univalence of two general integral operators”, Filomat, 29:7 (2015), 1581–1586 | DOI | MR | Zbl

[12] E.Deniz, “Convexity of integral operators involving generalized Bessel functions”, Integral Transforms Spec. Funct., 24 (2013), 201–216 | DOI | MR | Zbl

[13] E.Deniz, H.Orhan, H.M.Srivastava, “Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions”, Taiwan. J. Math., 15 (2011), 883–917 | MR | Zbl

[14] M.Din, H.M.Srivastava, M.Raza, “Univalence of integral operators involving generalized Struve functions”, Hacet. J. Math. Stat., 47 (2018), 821–833 | MR | Zbl

[15] M.U.Din, M.Raza, E.Deniz, “Univalence criteria for general integral operators involving normalized Dini functions”, Filomat, 34 (2020), 2203–2216 | DOI | MR | Zbl

[16] S.Eker, S.Ece, “Geometric Properties of the Rabotnov Functions”, Hacet. J. Math. Stat., 51:5 (2022), 1248–1259 | DOI | MR | Zbl

[17] B.A.Frasin, “Sufficient conditions for integral operator defined by Bessel functions”, J. Math. Ineq., 4 (2010), 301–306 | DOI | MR | Zbl

[18] B.A.Frasin, “Univalence criteria for general integral operator”, Math. Commun., 16 (2011), 115–124 | MR | Zbl

[19] B.A.Frasin, “New general integral operators of p-valent functions”, J. Ineq. Pure Appl.Math., 10:4 (2009), 109 | MR | Zbl

[20] B.A.Frasin, “Starlikeness and convexity of integral operators involving Mittag-Leffler functions”, TWMS J. App. Eng. Math., 14:3 (2024), 913–920

[21] B.A.Frasin, T.Al-Hawary, F.Yousef, “Some properties of a linear operator involving generalized Mittag-Leffler function”, Stud. Univ. Babeş-Bolyai Math., 65:1 (2020), 67–75 | DOI | MR | Zbl

[22] M.Garg, P.Manohar, S.L.Kalla, “A Mittag-Leffler-type function of two variables”, Integral Transforms Spec. Funct., 24:11 (2013), 934–944 | DOI | MR | Zbl

[23] S.Kazımoğlu, E.Deniz, “Partial sums of the Rabotnov function”, Acta Univ. Sapientiae, Mathematica, 14:2 (2022), 250–261 | DOI | MR | Zbl

[24] V.Kiryakova, “Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. Higher transcendental functions and their applications”, J. Comput. Appl. Math., 118:1-2 (2000), 241–259 | DOI | MR | Zbl

[25] V.Kiryakova, “The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus”, Comput. Math. Appl., 59:5 (2010), 1885–1895 | DOI | MR | Zbl

[26] F.Mainardia, R.Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes. Higher transcendental functions and their applications”, J. Comput. Appl. Math., 118:1-2 (2000), 283–299 | DOI | MR

[27] K.S.Miller, B.Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, New York–Chichester–Brisbane–Toronto–Singapore, 1993 | MR | Zbl

[28] G.M.Mittag-Leffler, “Sur la nouvelle fonction $E(x)$”, C. R. Acad. Sci. Paris, 137 (1903), 554–558

[29] J.H.Park, H.M.Srivastava, N.E.Cho, “Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind”, AIMS Math., 6 (2021), 11380–11402 | DOI | MR | Zbl

[30] N.Pascu, “An improvement of Becker's univalence criterion”, Proceedings of the Commemorative Session Simion Stoilow (Brasov, 1987), 43–98 | MR

[31] V.Pescar, “A new generalization of Ahlfor's and Becker's criterion of univalence”, Bull. Malaysian Math. Soc., 19:2 (1996), 53–54 | MR | Zbl

[32] V.Pescar, “Univalence of certain integral operators”, Acta Univ. Apulensis Math. Inform., 12 (2006), 43–48 | MR | Zbl

[33] Y.Rabotnov, “Equilibrium of an Elastic Medium with After-Effect”, Prikladnaya Matematika i Mekhanika, 12:1 (1948), 53–62 (in Russian) ; Reprinted: Fractional Calculus and Applied Analysis, 17:3 (2014), 684–696 | DOI | MR | Zbl | MR | Zbl

[34] M.Raza, M.U.Din, S.N.Malik, “Certain Geometric Properties of Normalized Wright Functions”, Journal of Function Spaces, 2016 (2016), 1896154 | MR | Zbl

[35] M.U.Din, M.Raza, S.Hussain, M.Darus, “Certain Geometric Properties of Generalized Dini Functions”, Journal of Function Spaces, 71 (2018), 1–9 | DOI | MR

[36] M.Raza, S.N.Malik, Q.Xin, M.U.Din, L.-I.Cotîrlă, “On Kudriasov Conditions for Univalence of Integral Operators Defined by Generalized Bessel Functions”, Mathematics, 10 (2022), 1361 | DOI

[37] H.M.Srivastava, Z.Tomovski, “Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel”, Appl. Math. Comp., 211 (2009), 198–210 | DOI | MR | Zbl

[38] H.M.Srivastava, B.A.Frasin, V.Pescar, “Univalence of integral operators involving Mittag–Leffler functions”, Appl. Math. Inf. Sci., 11 (2017), 635–641 | DOI | MR

[39] N.Ularu, “The univalence of some integral operators using the Bessel functions”, Math. Vasnik, 65 (2013), 547–554 | MR | Zbl

[40] A.Wiman, “Über den Fundamental satz in der Theorie der Funcktionen $\mathbb{R}(x)$”, Acta Math., 29 (1905), 191–201 | DOI | MR

[41] A.Wiman, “Über die Nullstellun der Funcktionen $E(x)$”, Acta Math., 29 (1905), 217–134 | DOI | MR

[42] N.Yagmur, H.Orhan, “Hardy space of generalized Struve functions”, Complex Variables and Elliptic Equations, 59:7 (2014), 929–936 | DOI | MR | Zbl