Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ON A CLASS OF QUANTUM ALPHA-CONVEX FUNCTIONS

  • NOOR, KHALIDA INAYAT (Department of Mathematics, COMSATS University Islamabad) ;
  • BADAR, RIZWAN S. (Department of Mathematics, COMSATS University Islamabad)
  • Received : 2018.04.19
  • Accepted : 2018.07.02
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let $f:f(z)=z+{\sum^{{\infty}}_{n=2}}a_nz^n$ be analytic in the open unit disc E. Then f is said to belong to the class $M_{\alpha}$ of alpha-convex functions, if it satisfies the condition ${\Re}\{(1-{{\alpha})}{\frac{zf^{\prime}(z)}{f(z)}}+{{\alpha}}{\frac{(zf^{\prime}(z))^{\prime})}{f^{\prime}(z)}}\}$ > 0, ($z{\in}E$). In this paper, we introduce and study q-analogue of the class $M_{\alpha}$ by using concepts of Quantum Analysis. It is shown that the functions in this new class $M(q,{\alpha})$ are q-starlike. A problem related to q-Bernardi operator is also investigated.

Keywords

References

  1. K. Ademogullari and Y. Kahramaner, q-harmonic mappings for which analytic part is q-convex function, Nonlinear Anal. Diff. Eqns. 4(2016), 283-293.
  2. M.H. Ismail, E. Merkes and D. styer, A generalization of starlike functions, Complex Var. Elliptic Eqns. 14(1990), 77-84.
  3. F.H. Jackson, On q-functions and certain difference operators, Trans. Roy. Soc. Edinburgh 46(1909), 253-281.
  4. F.H. Jackson, On q-definite integrals, Q. J. Math. 41(1910), 193-203.
  5. S.S. Miller, P.T. Mocanu and M.O. Reade, All ${\alpha}$-convex functions are starlike, Proc. Amer. Math. Soc. 37(1973), 553-554.
  6. A. Muhammad and M. Darus, A generalized operator involving the q-hyperbolic functions, Mat. Vesnik 65(2013), 454-465.
  7. K.I. Noor, On generalized q-close-to-convexity, Appl. Math. Inform. Sci. 11(5) (2017), 13831388 https://doi.org/10.18576/amis/110515
  8. K.I. Noor, On generalized q-Bazilevic functions, J. Adv. Math. Stud. 10(2017), 418-424.
  9. K.I. Noor and S. Riaz, Generalized q-starlike functions, Studia Sci. Hungar. 54(4)(2017), 509-522.
  10. K.I. Noor, S. Riaz and M.A. Noor, On q-Bernardi integral opertaor, TWMS J. Pure Appl, Math. 8(1)(2017), 3-11.
  11. K.I. Noor and M.A. Noor, Linear combinations of generalized q-starlike functions, Appl. Math. Info. Sci. 11(2017), 745-748. https://doi.org/10.18576/amis/110314
  12. S.K. Sahoo and N.L. Sharma, On a generalization of close-to-convex functions, arXiv:1404.3268 [math. CV], 14 pp. https://doi.org/10.4064/ap113-1-6
  13. H.E.O. Ucar, Coefficeient inequality for q-starlike functions, Appl. Math. Comput. 276(2016), 122-126.