Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On the Fekete-Szegö Problem for a Certain Class of Meromorphic Functions Using q-Derivative Operator

  • Aouf, Mohamed Kamal (Department of Mathematics, Faculty of Science, Mansoura University) ;
  • Orhan, Halit (Department of Mathematics, Faculty of Science, Ataturk University)
  • Received : 2017.10.06
  • Accepted : 2018.03.29
  • Published : 2018.06.23
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we obtain $Fekete-Szeg{\ddot{o}}$ inequalities for certain class of meromorphic functions f(z) for which $-{\frac{(1-{\frac{{\alpha}}{q}})qzD_qf(z)+{\alpha}qzD_q[zD_qf(z)]}{(1-{\frac{{\alpha}}{q}})f(z)+{\alpha}zD_qf(z)}{\prec}{\varphi}(z)$(${\alpha}{\in}{\mathbb{C}}{\backslash}(0,1]$, 0 < q < 1). Sharp bounds for the $Fekete-Szeg{\ddot{o}}$ functional ${\mid}{\alpha}_1-{\mu}{\alpha}^2_0{\mid}$ are obtained.

Keywords

References

  1. M. H. Abu-Risha, M. H. Annaby, M. E. H. Ismail and Z. S. Mansour, Linear q-difference equations, Z. Anal. Anwend., 26(2007), 481-494.
  2. R. M. Ali and V. Ravichandran, Classes of meromorphic ${\alpha}-$convex functions, Taiwanese J. Math., 14(4)(2010), 1479-1490. https://doi.org/10.11650/twjm/1500405962
  3. M. K. Aouf, Coefficient results for some classes of meromorphic functions, J. Natur. Sci. Math., 27(2)(1987), 81-97.
  4. M. K. Aouf, R. M. El-Ashwah and H. M. Zayed, Fekete-Szego inequalities for certain class of meromorphic functions, J. Egyptian Math. Soc., 21(2013), 197-200. https://doi.org/10.1016/j.joems.2013.03.013
  5. M. K. Aouf, A. O. Mostafa and,H. M. Zayed, Convolution properties for some subclasses of meromorphic functions of complex order, Abstr. Appl. Anal., Art. ID 973613 (2015), 6 pp.
  6. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
  7. M. E. H. Ismail, E. Merkes and D. Styer, A generalization of starlike functions, Complex Variables Theory Appl., 14(1990), 77-84. https://doi.org/10.1080/17476939008814407
  8. F. H. Jackson, The application of basic numbers to Bessel's and Legendre's functions, (Second paper), Proc. London Math. Soc., 3(2)(1904-1905), 1-23.
  9. V. G. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002.
  10. S. Kanas and D. Raducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64(5)(2014) 1183-1196. https://doi.org/10.2478/s12175-014-0268-9
  11. V. Karunakaran, On a class of meromorphic starlike functions in the unit disc, Math. Chronicle, 4(2-3)(1976), 112-121.
  12. S. R. Kulkarni and S. S. Joshi, On a subclass of meromorphic univalent functions with positive coefficients, J. Indian Acad. Math., 24(1)(2002), 197-205.
  13. W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994, 157-169.
  14. J. E. Miller, Convex meromorphic mappings and related functions, Proc. Amer. Math. Soc., 25(1970), 220-228. https://doi.org/10.1090/S0002-9939-1970-0259098-7
  15. Z. Nehari, Conformal Mapping, McGraw-Hill, New York, 1952.
  16. Ch. Pommerenke, On meromorphic starlike functions, Pacific J. Math., 13(1963), 221-235. https://doi.org/10.2140/pjm.1963.13.221
  17. T. Shanmugam and S. Sivasubramanian, On the Fekete-Szego problem for some subclasses of analytic functions, J. Inequal. Pure Appl. Math., 6(3)(2005), Article 71, 6pp.
  18. H. Silverman, K. Suchithra, B. A. Stephen and A. Gangadharan, Coefficient bounds for certain classes of meromorphic functions, J. Inequal. Appl., (2008), Art. ID 931981, 9 pp.
  19. S. Sivasubramanian and M.Govindaraj, On a class of analytic functions related to conic domains involving q-calculus, Anal. Math., 43(3)(2017), 475-487. https://doi.org/10.1007/s10476-017-0206-5