Partial sums and neighborhoods of Janowski-type subclasses of meromorphic functions
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Abstract
The paper presents the introduction of a novel linear derivative operator for meromorphic functions that are linked with $q$-calculus. Using the linear derivative operator, a new category of meromorphic functions is generated in the paper. We obtain sufficient conditions and show some properties of functions belonging to these subclasses. The partial sums of its sequence and the $q$-neighborhoods problem are solved.
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References
[1] A. Alatawi, M. Darus and S. Sivasubramanian, Generalised subclasses of meromorphically q- starlike function using the Janowski functions, Mathematical Foundations of Computing, 2023. https://doi.org/10.3934/mfc.2023021. Google Scholar
[2] A. Alatawi, M. Darus, B. Alamri, Applications of Gegenbauer polynomials for subfamilies of bi-univalent functions involving a Borel distribution-type Mittag-Leffler function, Symmetry. 15 (2023), 785. Google Scholar
[3] O. Altintas and S. Owa, Neighborhoods of cartain analytic functions with negative coefficients, International Journal of Mathematics and Mathematical Sciences. 19 (1996), 797–800. Google Scholar
[4] J. Clune, On meromorphic Schlicht functions, Journal of the London Mathematical Society. 1 (1959), 215–216. https://doi.org/10.1112/jlms/s1-34.2.215 Google Scholar
[5] M. Darus and R.W. Ibrahim, On partial sums of generalized differential operator, Proc, Pakistan Acad. Sci. 46 (2009), 209–215. Google Scholar
[6] S. Elhaddad, H. Aldweby and M. Darus, Neighborhoods of certain classes of analytic functions defined by a generalized differential operator involving Mittag-Leffler function, Acta Universitatis Apulensis. 18 (2018), 1–10. Google Scholar
[7] G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge university press, 2004. Google Scholar
[8] A.W. Goodman, Univalent functions and non analytic curves, Proc. Amer. Math. Soc. 8 (1957), 598–601. Google Scholar
[9] S.H. Hadi, M. Darus and A. Alb Lupa ̧s, A class of Janowski-type (p, q)-convex harmonic functions involving a generalized q-Mittag–Leffler function, Axioms. 12 (2023), 190. Google Scholar
[10] F.H. Jackson, On q-functions and a certain difference operator, Transactions of the Royal Society of Edinburgh. 46 (1909), 253–281. Google Scholar
[11] F.H. Jackson, On q-definite integrals, The Quarterly Journal of Pure and Applied Mathematics. 41 (1910), 193–203. Google Scholar
[12] W. Janowski, Some extremal problems for certain families of analytic functions, Annales Polonici Mathematici. 28 (1973), 297–326. Google Scholar
[13] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002. Google Scholar
[14] V. Karunakaran, On a class of meromorphic starlike functions in the unit disc, Mathematical chronicle. 4 (1974),112–121. Google Scholar
[15] G. Murugusundaramoorthy and S.V.S. Velayudam, Neighborhoods and Partial sums of meromorphic Univalent Functions, Mapana Journal of Sciences. 4 (2005), 22–31. Google Scholar
[16] J.E. Miller, Convex meromorphic mappings and related functions, Proceedings of the American Mathematical Society. 25 (1970), 220–228. Google Scholar
[17] A. Mohammed and M. Darus, A generalized operator involving the q-hypergeometric function, Matematicki Vesnik. 65 (2013), 454–465. Google Scholar
[18] S. Mahmood, Q.Z. Ahmad, H.M. Srivastava, N. Khan, B. Khan, and M. Tahir, A certain subclass of meromorphically q-starlike functions associated with the Janowski functions, Journal of Inequalities and Applications. (2019), 01–11. Google Scholar
[19] L.M. Morga, Meromorphic multivalent functions with positive coefficients, Mathematica Japonica. 35 (1990), 01–11. Google Scholar
[20] C. Pommerenke, On meromorphic starlike functions, Pacific Journal of Mathematics. 13 (1963), 221–235. Google Scholar
[21] W.C. Royster, Meromorphic starlike multivalent functions, Transactions of the American Mathematical Society. 107 (1963), 300–308. Google Scholar
[22] T.M. Seoudy and M.K. Aouf, Coefficient estimates of new classes of q-starlike and q-convex functions of complex order, Jornal of Mathematical Inequalities. 10 (2016), 135–145. Google Scholar
[23] A. S. Shinde, R. N. Ingle and P. T. Reddy, On Certain Subclass Of Meromorphic Functions With Positive Coefficients, Palestine Journal of Mathematics. 10 (2021),685–693. Google Scholar
[24] H.M. Srivastava, H.M. Hossen and M.K. Aouf, A unified presentation of some classes of mero- morphically multivalent functions, Comput. Math. Appl. 38 (1996), 63–70. Google Scholar
[25] H. Silverman, Neighborhoods of a classes of analytic function, Far East J. Math.Sci. 3 (1995), 165–169. Google Scholar
[26] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl. 209 (1997), 221–227. Google Scholar
[27] H.M. Srivastava, S.H. Hadi and M. Darus, Some subclasses of p-valent γ-uniformly type q-starlike and q-convex functions defined by using a certain generalized q-Bernardi integral operator, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117 (2023), 65. Google Scholar
[28] H. Tang, H. Zayed and A. Mostafa and M. Aouf, Fekete-Szego Problems for Certain Classes of Meromorphic Functions Using q-Derivative Operator, Journal of Mathematical Research with Applications. 38 (3) (2018), 236–246. http://dx.doi.org/10.3770/j.issn:2095-2651.2018.03.002 Google Scholar