Research article

Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator

  • Received: 20 October 2017 Accepted: 31 October 2017 Published: 16 November 2017
  • MSC : Primary 30C45; Secondary 30C50

  • In our present investigation, by using Salagean q-differential operator we introduce and define new subclass $k-\mathcal{US}(q, \gamma, m), $ $\gamma \in C\backslash \{0\}, $ and studied certain subclass of analytic functions in conic domains. We investigate the number of useful properties of this class such structural formula and coefficient estimates Fekete--Szego problem, we give some subordination results, and some other corollaries.

    Citation: Saqib Hussain, Shahid Khan, Muhammad Asad Zaighum, Maslina Darus. Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator[J]. AIMS Mathematics, 2017, 2(4): 622-634. doi: 10.3934/Math.2017.4.622

    Related Papers:

  • In our present investigation, by using Salagean q-differential operator we introduce and define new subclass $k-\mathcal{US}(q, \gamma, m), $ $\gamma \in C\backslash \{0\}, $ and studied certain subclass of analytic functions in conic domains. We investigate the number of useful properties of this class such structural formula and coefficient estimates Fekete--Szego problem, we give some subordination results, and some other corollaries.


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    [1] W. Ma, D. Minda, Uniformly convex functions, Ann. Polon. Math., 57 (1992), 165-175.
    [2] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Am. Math. Soc., 118 (1993), 189-196.
    [3] A. W Goodman, Univalent Functions, vols. Ⅰ, Ⅱ, Polygonal Publishing House, New Jersey, 1983.
    [4] A. W Goodman, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87-92.
    [5] S. Kanas, A.Wisniowska, Conic domains and k-starlike functions, Rev. Roum. Math. Pure Appl., 45 (2000), 647-657.
    [6] S. Kanas, A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105 (1999), 327-336.
    [7] K.G Subramanian, G. Murugusundaramoorthy, P.Balasubrahmanyam, H. Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math. Jpn., 42 (1995), 517-522.
    [8] R. Bharati, R. Parvatham, A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math., 28 (1997), 17-32.
    [9] H. S. Al-Amiri, T. S. Fernando, On close-to-convex functions of complex order, Int. J. Math. Math. Sci., 13 (1990), 321-330.
    [10] M. Acu, Some subclasses of α-uniformly convex functions, Acta Math. Acad. Pedagogicae Nyiregyhaziensis, 21 (2005), 49-54.
    [11] A. Gangadharan, T. N Shanmugam, H. M., Srivastava, Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. Math. Appl., 44 (2002), 1515-1526.
    [12] A. Swaminathan, Hypergeometric functions in the parabolic domain, Tamsui Oxf. J. Math. Sci., 20 (2004), 1-16.
    [13] S. Kanas, Techniques of the differential subordination for domain bounded by conic sections, Int. J. Math. Math. Sci., 38 (2003), 2389-2400.
    [14] N. Khan, B. Khan, Q. Z. Ahmad and S. Ahmad, Some Convolution Properties of Multivalent Analytic Functions, AIMS Math., 2 (2017), 260-268.
    [15] S. S. Miller, P. T. Mocanu, Differential Subordinations: Theory and Applications, Series of Monographs and Textbooks in Pure and Application Mathematics, vol. 225. Marcel Dekker, New York, 2000.
    [16] S. Kanas, D. Raducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183-1196.
    [17] S. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc., 49 (1975), 109-115.
    [18] K. I..Noor, M. Arif, W. Ul-Haq, On k-uniformly close-to-convex functions of complex order, Appl. Math. Comput., 215 (2009), 629-635.
    [19] W. Rogosinski, On the coeffcients of subordinate functions, Proc. Lond. Math. Soc., 48 (1943), 48-82.
    [20] S. J. Sim, O. S., Kwon, N. E. Cho, H. M. Srivastava, Some classes of analytic functions associated with conic regions, Taiwan. J. Math., 16 (2012), 387-408.
    [21] W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, Z. Li, F. Ren, L. Yang, S. Zhang (Eds. ) pp. 157-169, International Press, Cambridge, MA, 1994.
    [22] Z. Shareef, S. Hussain, M. Darus, Convolution operator in geometric functions theory, J. Inequal. Appl., 2012,2012:213.
    [23] K. I. Noor, M. A Noor, On certain classes of analytic functions defined by Noor integral operator, J. Math. Anal. Appl., 281 (2003), 244-252.
    [24] S. Mahmood, J. Sokol, New subclass of analytic functions in conical domain associated with ruscheweyh q-Differential operator, Results Math., 71 (2017), 1345-1357.
    [25] S. Shams, S. R. Kulkarni, J. M. Jahangiri, Classes of uniformly starlike and convex functions, Int. J. Math. Math. Sci., 55 (2004), 2959-2961.
    [26] H. Selverman, Univalent functions with negative coeffcients, Proc. Amer. Math. Soc., 51 (1975), 109-116.
    [27] S. Owa, Y. Polatoglu, E.Yavuz, Coeffcient inequalities for classes of uniformly starlike and convex functions, J. Ineq. Pure Appl. Math., 7 (2006), 1-5.
    [28] R. M. Ali, Starlikeness associated with parabolic regions, Int. J. Math. Sci., 4 (2005), 561-570.
    [29] M. Govindaraj and S. Sivasubramanian, On a class of analytic functions related to conic domains involving q-calculus, Analysis Math., 43 (2017), 475-487.
    [30] G. S. Salagean, Subclasses of univalent functions, in: Complex Analysis, fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Mathematics, 1013, Springer (Berlin, 1983), 362-372.
    [31] G. E. Andrews, G. E. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
    [32] C. R. Adams, On the linear partial q-difference equation of general type, Trans. Amer. Math. Soc., 31 (1929), 360-371.
    [33] R. D. Carmichael, The general theory of linear q-difference equations, Amer. J. Math., 34 (1912), 147-168.
    [34] F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
    [35] T. E. Mason, On properties of the solution of linear q-difference equations with entire function coeffcients, Amer. J. Math., 37 (1915), 439-444.
    [36] W. J. Trjitzinsky, Analytic theory of linear q-difference equations, Acta Math., 61 (1933), 1-38.
    [37] M. E. H. Ismail, E. Merkes and D. Styer, A generalization of starlike functions, Complex Variables Theory and Appl., 14 (1990), 77-84.
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