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A subclass of analytic functions defined by using Mittag-Leffler function. (English) Zbl 1467.30010

Summary: In this paper, new subclasses of analytic functions are proposed by using Mittag-Leffler function. Also some properties of these classes are studied in regard to coefficient inequality, distortion theorems, extreme points, radii of starlikeness and convexity and obtained numerous sharp results.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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[1] A. Abubakar and M. Darus, On a certain subclass of analytic functions involving differential operators, Transyl. J. Math. Mech. 3 (2011), 1-8. · Zbl 1243.30009
[2] M. K. Aouf, A subclass of uniformly convex functions with negative coefficients, Mathematica 52 (2010), 99-111. · Zbl 1224.30026
[3] A. A. Attiya, Some applications of Mittage-Leffler function in the unit disc, Filomat 30 (2016), 2075-2081. · Zbl 1458.30006 · doi:10.2298/FIL1607075A
[4] R. Bharati, R. Parvatham and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28 (1997), 17-32. · Zbl 0898.30010
[5] M. Darus, S. Hussain, M. Raza and J. Sokol, On a subclass of starlike functions, Results Math. 73 (2018), 1-12. · Zbl 1390.41009 · doi:10.1007/s00025-018-0773-1
[6] S. Elhaddad, M. Darus and H. Aldweby, On certain subclasses of analytic functions involving differential operator, Jnanabha 48 (2018), 53-62. · Zbl 1411.30011
[7] I. Faisal and M. Darus, Study on subclass of analytic functions, Acta Univ. Sapientiae Mathematica 9 (2017), 122-139. · Zbl 1372.30007 · doi:10.1515/ausm-2017-0008
[8] A. Fernandez, D. Baleanu and H. M. Srivastava, Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simulat. 67 (2019), 517-527. · Zbl 1508.26006 · doi:10.1016/j.cnsns.2018.07.035
[9] A. W. Goodman, Univalent Functions, vols. I, II. Polygonal Publishing House, New Jersey, 1983. · Zbl 1041.30501
[10] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. · Zbl 0744.30010 · doi:10.4064/ap-56-1-87-92
[11] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28 (1973), 648-658. · Zbl 0275.30009
[12] S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), 327-336. · Zbl 0944.30008 · doi:10.1016/S0377-0427(99)00018-7
[13] S. Kanas and A. Wisniowska, Conic domains and k-starlike functions, Rev. Roum. Math. Pure Appl. 45 (2000), 647-657. · Zbl 0990.30010
[14] S. Kanas, S. Altinkaya and S. Yalcin, Subclass of k-uniformly starlike functions defined by symmetric q-derivative operator, Ukrainian Mathematical Journal 70 (2019), 1499-1510. · Zbl 1435.30048
[15] S. Kanas and S. Altinkaya, Functions of bounded variation related to domains bounded by conic sections, Mathematica Slovaca 69 (2019), 833-842. · Zbl 1498.30006 · doi:10.1515/ms-2017-0272
[16] J. E. Littlewood, On inequalities in the theory of functions, Proceedings of the London Mathematical Society 23 (1925), 481-519.
[17] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. · Zbl 0760.30004 · doi:10.4064/ap-57-2-165-175
[18] S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and Applications, Series of Monographs and Textbooks in Pure and Application Mathematics vol. 255. Dekker, New York, 2000. · Zbl 0954.34003
[19] G. M. Mittag-Leffler, Sur la nouvelle function \(E_{\bar{\alpha}}(x)\), C. R. Acad. Sci. Paris 137 (1903), 554-558.
[20] G. M. Mittag-Leffler, Sur la representation analytique d’une branche uniform d’une function monogene, Acta Mathematica 29 (1905), 101-181. · doi:10.1007/BF02403200
[21] K. I. Noor and S. Hussain, On certain analytic functions associated with Ruscheweyh derivatives and bounded Mocanu variation, J. Math. Anal. Appl. 340 (2008), 1145-1152. · Zbl 1155.30007 · doi:10.1016/j.jmaa.2007.09.038
[22] K. S. Padmanabhan and M. S. Ganesan, Convolutions of certain classes of univalent functions with negative coefficients, Indian Journal of Pure and Applied Mathematics 19 (1988), 880-889. · Zbl 0652.30009
[23] F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Am. Math. Soc. 118 (1993), 118: 189-196. · Zbl 0805.30012 · doi:10.1090/S0002-9939-1993-1128729-7
[24] H. Rehman, M. Darus and J. Salah, Coefficient properties involving the generalized k-Mittag-Leffler functions, Transyl. J. Math. Mech. 9 (2017), 155-164.
[25] S. Shams, S. R. Kulkarni and J. M. Jahangiri, On a class of univalent functions defined by Ruschweyh derivatives, Kyungpook Mathematical Journal 43 (2003), 579-585. · Zbl 1067.30032
[26] S. Shams, S. R. Kulkarni and J. M. Jahangiri, Classes of uniformly starlike and convex functions, International Journal of Mathematics and Mathematical Sciences 53 (2004), 2959-2961. · Zbl 1067.30033
[27] H. Silverman, Univalent functions with negative coefficients, Proceedings of the American Mathematical Society 51 (1975), 109-116. · Zbl 0311.30007 · doi:10.1090/S0002-9939-1975-0369678-0
[28] H. M. Srivastava, B. A. Frasin and V. Pescar, Univalence of integral operators involving Mittag-Leffler functions, Appl. Math. Inf. Sci. 11 (2017), 635-641. · doi:10.18576/amis/110301
[29] H. M. Srivastava, A. R. S. Juma and H. M. Zayed, Univalence conditions for an integral operator defined by a generalization of the Srivastava-Attiya operator, Filomat 32 (2018), 2101-2114. · Zbl 1499.30171 · doi:10.2298/FIL1806101S
[30] H. M. Srivastava and H. Gunerhan, Analytical and approximate solutions of fractional-order susceptible-infected-recovered epidemic model of childhood disease, Math. Methods Appl. Sci. 42 (2019), 935-941. · Zbl 1410.34140 · doi:10.1002/mma.5396
[31] K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam and H. Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math. Jpn. 42 (1995), 517-522. · Zbl 0837.30011
[32] S. Sumer Eker and S. Owa, Certain classes of analytic functions involving Salagean operator, Journal of Inequalities in Pure and Applied Mathematics 10 (2009), 12-22. · Zbl 1165.26325
[33] A. Wiman, Uber den fundamentalsatz in der teorie der funktionen \(E_{\bar{\alpha}}(x)\), Acta Mathematica 29 (1905), 191-201. · doi:10.1007/BF02403202
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