\(p\)-Valent strongly starlike and strongly convex functions connected with linear differential Borel operator

Volume 26, Issue 2, pp 137--148 http://dx.doi.org/10.22436/jmcs.026.02.04

Publication Date: November 05, 2021 Submission Date: July 30, 2021 Revision Date: September 02, 2021 Accteptance Date: September 17, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1322 Downloads
• 2956 Views

Authors

S. M. El-Deeb - Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt. - Department of Mathematics, College of Science and Arts, Al-Badaya, Qassim University, Buraidah 51452, Saudi Arabia. G. Murugusundaramoorthy - Department of Mathematics, School of Advanced Sciences, Vellore Institute Technology University, Vellore - 632014, India. A. Alburaikan - Department of Mathematics, College of Science and Arts, Al-Badaya, Qassim University, Buraidah 51452, Saudi Arabia.

Abstract

In this paper, we define two subclasses \(\mathcal{S}^{\ast }\left( \alpha ,m,\delta ,p,q,\lambda ,\gamma ,\beta \right) \) and \(\mathcal{K}\left( \alpha ,m,\delta ,p,q,\lambda ,\gamma ,\beta \right) \) of strongly starlike and strongly convex functions of order \(\beta \) and type \(\gamma \ \)by using the linear \(q\)-differential Borel operator. We also derive some interesting properties, such as inclusion relationships of these classes and the integral operator \(\mathcal{J}_{\mu ,p}\).

Share and Cite

Keywords

• \(p\)-Valent
• strongly starlike
• strongly convex
• linear \(q\)-differential Borel operator

MSC

•  05A30
•  30C45
•  11B65
•  47B38

References

• [1] M. H. Abu Risha, M. H. Annaby, H. E. H. Ismail, Z. S. Mansour, Linear $q$-difference equations, Z. Anal. Anwend., 26 (2007), 481--494
  • View Article
  • MathSciNet
  • Google Scholar


• [2] M. K. Aouf, S. M. El-Deeb, On $p$-valent strongly starlike and strongly convex functions assoicated with generalized linear operator, Montes Taurus J. Pure Appl. Math., 3 (2021), 104--111
  • View Article
  • Google Scholar


• [3] M. K. Aouf, H. M. Hossen, New criteria for meromorphic $p$-valent starlike functions, Tsukuba J. Math., 17 (1993), 481--486
  • View Article
  • MathSciNet
  • Google Scholar


• [4] J. H. Choi, M. Saigo, H. M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl., 276 (2002), 432--445
  • View Article
  • MathSciNet
  • Google Scholar


• [5] S. M. El-Deeb, B. M. El-Matary, Coefficients Bounds of Multivalent Function Connected with $q$-Analogue of Salagean Operator, Appl. Math. Inf. Sci., 14 (2020), 1057--1065
  • View Article


• [6] S. M. El-Deeb, G. Murugusundaramoorthy, Applications on a subclass of $\beta$-uniformly starlike functions connected with $q$-Borel distribution, J. Interdisciplinary Math., (Submitted)


• [7] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990)
  • View Article


• [8] A. W. Goodman, On the Schwarz Christoffel transformation and $p$-valent functions, Trans. Amer. Math. Soc., 68 (1950), 204--233
  • View Article
  • MathSciNet
  • Google Scholar


• [9] F. H. Jackson, On $q$-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46 (1909), 253--281
  • View Article


• [10] D. O. Jackson, T. Fukuda, O. Dunn, E. Majors, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193--203
  • View Article
  • Google Scholar


• [11] N. Khan, Q. Z. Ahmad, T. Khalid, B. Khan, Results on spirallike $p$-valent functions, AIMS Math., 1 (2017), 12--20
  • View Article
  • Google Scholar


• [12] B. Khan, Z.-G. Liu, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, A study of some families of multivalent $q$-starlike functions involving higher-order $q$-Derivatives, Mathematics, 8 (2020), 12 pages
  • View Article
  • Google Scholar


• [13] A. Muhammad, On $p$-valent strong ly starlike and strongly convex functions, Acta Univ. Apulensis, 26 (2011), 179--188
  • View Article
  • Google Scholar


• [14] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 234--237
  • View Article
  • MathSciNet
  • Google Scholar


• [15] M. Nunokawa, S. Owa, A. Ikeda, On the strongly starlikeness of multivalently convex functions of order $\alpha$, Int. J. Math. Math. Sci., 28 (2001), 51--55
  • View Article
  • MathSciNet
  • Google Scholar


• [16] M. Nunokawa, S. Owa, H. Saitoh, A. Ikeda, N. Koike, Some results for strongly starlike functions, J. Math. Anal. Appl., 212 (1997), 98--106
  • View Article
  • MathSciNet
  • Google Scholar


• [17] M. Obradovc, S. B. Joshi, On certain classes of strongly starlike functions, Taiwan. J. Math., 2 (1998), 297--302
  • View Article
  • MathSciNet
  • Google Scholar


• [18] S. Owa, On certain classes of p-valent functions with negative coefficients, Simon Stevin, 59 (1985), 385--402
  • MathSciNet
  • Google Scholar


• [19] D. A. Patel, N. K. Thakare, On convex hulls and extreme points of $p$-valent starlike and convex classes with applications, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 27 (1983), 145--160
  • View Article
  • MathSciNet
  • Google Scholar


• [20] J. K. Prajapat, S. P. Goyal, Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal., 3 (2009), 129--137
  • View Article
  • MathSciNet
  • Google Scholar


• [21] H. M. Srivastava, Operators of basic (or $q$-) calculus and fractional $q$-calculus and their applications in Geometric Function theory of Complex Analysis, Iran J. Sci. Technol. Trans. Sci., 44 (2020), 327--344
  • View Article
  • Google Scholar


• [22] H. M. Srivastava, Q. Z. Ahmad, N. Khan, S. Kiran, B. Khan, Some applications of higher order derivatives involving certain subclasses of analytic and multivalent functions, J. Nonlinear Var. Anal., 2 (2018), 343--353
  • View Article
  • Google Scholar


• [23] H. M. Srivastava, S. M. El-Deeb, A certain class of analytic functions of complex order connected with a $q$-analogue of integral operators, Miskolc Math. Notes, 21 (2020), 417--433
  • View Article
  • MathSciNet
  • Google Scholar


• [24] A. K. Wanas, J. A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline J. Math. Sci., 4 (2020), 71--82
  • View Article
  • Google Scholar