Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Univalent Functions Associated with the Symmetric Points and Cardioid-shaped Domain Involving (p,q)-calculus

  • Ahuja, Om (Department of Mathematical Sciences, Kent State University) ;
  • Bohra, Nisha (Department of Mathematics, Sri Venkateswara College, University of Delhi) ;
  • Cetinkaya, Asena (Department of Mathematics and Computer Science, Istanbul Kultur University) ;
  • Kumar, Sushil (Bharati Vidyapeeth's college of Engineering)
  • Received : 2020.03.19
  • Accepted : 2020.07.28
  • Published : 2021.03.31
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Abstract

In this paper, we introduce new classes of post-quantum or (p, q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p, q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p, q)-logarithmic coefficients. In addition, we get q-Bieberbachde-Branges type inequalities for the special case of our classes when p = 1. Moreover, we also discuss some special cases of the obtained results.

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References

  1. O. P. Ahuja, Hadamard products and neighbourhoods of spirallike functions, Yokohama Math. J., 40(2)(1993), 95-103.
  2. O. P. Ahuja and A. Cetinkaya, Use of quantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conference Proceedings 2095, 020001 (2019); https://doi.org/10.1063/1.5097511.
  3. O. P. Ahuja, A. Cetinkaya and Y. Polatoglu, Bieberbach-de Branges and Fekete-Szego inequalities for certain families of q-convex and q-close to convex functions, J. Comput. Anal. Appl., 26(4)(2019), 639-649.
  4. M. F. Ali and A. Vasudevarao, On logarithmic coefficients of some close-to-convex functions, Proc. Amer. Math. Soc., 146(3)(2018), 1131-1142. https://doi.org/10.1090/proc/13817
  5. A. Aral, V. Gupta and R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, 2013.
  6. R. Bucur and D. Breaz, On a new class of analytic functions with respect to symmetric points involving the q-derivative operator, J. Phys.: Conf. Series 1212 (2019), doi:10.1088/1742-6596/1212/1/012011.
  7. A. Cetinkaya, Y. Kahramaner and Y. Polatoglu, Fekete-Szego inequalities for q-starlike and q-convex functions, Acta Univ. Apulensis Math. Inform., 53(2018), 55-64.
  8. R. Chakrabarti and R. Jagannathan, A (p, q)-oscillator realization of two-parameter quantum algebras, J. Phys. A, 24(13)(1991), L711-L718. https://doi.org/10.1088/0305-4470/24/13/002
  9. N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko and Y. J. Sim, On the third logarithmic coefficient in some subclasses of close-to-convex functions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(2)(2020), Paper No. 52, 14 pp.
  10. R. N. Das and P. Singh, On subclasses of schlicht mapping, Indian J. Pure Appl. Math., 8(8)(1977), 864-872.
  11. P. L. Duren and Y. J. Leung, Logarithmic coefficients of univalent functions, J. Analyse Math., 36(1979), 36-43. https://doi.org/10.1007/BF02798766
  12. M. Fekete and G. Szego, Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc., 8(2)(1933), 85-89.
  13. M. E. H. Ismail, E. Merkes and D. Styer, A generalization of starlike functions, Complex Variables Theory Appl., 14(1-4) (1990), 77-84. https://doi.org/10.1080/17476939008814407
  14. F. H. Jackson, On q-functions and certain difference operators, Transactions of the Royal Society of Edinburgh, 46(1908), 253-281. https://doi.org/10.1017/S0080456800002751
  15. F. H. Jackson, On q-definite integrals, Quar. J. Pure Appl. Math., 41(1910), 193-203.
  16. B. Kowalczyk, A. Lecko and H. M. Srivastava, A note on the Fekete-Szego problem for close-to-convex functions with respect to convex functions, Publ. Inst. Math. (Beograd) (N.S.), 101(115)(2017), 143-149. https://doi.org/10.2298/PIM1715143K
  17. S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math., 40(2)(2016), 199-212.
  18. S. Kumar and V. Ravichandran, Functions defined by coefficient inequalities, Malays. J. Math. Sci., 11(3)(2017), 365-375.
  19. S. Kumar, V. Ravichandran and S. Verma, Initial coefficients of starlike functions with real coefficients, Bull. Iranian Math. Soc., 43(6)(2017), 1837-1854.
  20. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157-169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994.
  21. H. E. Ozkan Ucar, Coefficient inequality for q-starlike functions, Appl. Math. Comput., 276(2016), 122-126. https://doi.org/10.1016/j.amc.2015.12.008
  22. U. Pranav Kumar and A. Vasudevarao, Logarithmic coefficients for certain subclasses of close-to-convex functions, Monatsh. Math., 187(3)(2018), 543-563. https://doi.org/10.1007/s00605-017-1092-4
  23. D. V. Prokhorov and J. Szynal, Inverse coefficients for (α, β)-convex functions, Ann. Univ. Mariae Curie-Sk lodowska Sect. A, 35(1981), 125-143.
  24. C. Ramachandran, D. Kavitha and T. Soupramanien, Certain bound for q-starlike and q-convex functions with respect to symmetric points, Int. J. Math. Math. Sci., (2015), Art. ID 205682, 7 pp.
  25. C. Ramachandran, D. Kavitha and W. Ul-Haq, Fekete Szego theorem for a close-to-convex error function, Math. Slovaca, 69(2)(2019), 391-398. https://doi.org/10.1515/ms-2017-0231
  26. K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11(1959), 72-75. https://doi.org/10.2969/jmsj/01110072
  27. C. Selvaraj and N. Vasanthi, Subclasses of analytic functions with respect to symmetric and conjugate points, Tamkang J. Math., 42(1)(2011), 87-94. https://doi.org/10.5556/j.tkjm.42.2011.868
  28. S. Sivasubramanian and P. Gurusamy, The Fekete-Szego coefficient functional problems for q-starlike and q-convex functions related with lemniscate of Bernoulli, Asian-Eur. J. Math., 12(2) (2019), 1950019, 14 pp. https://doi.org/10.1142/s1793557119500190
  29. J. Soko land D. K. Thomas, Further results on a class of starlike functions related to the Bernoulli lemniscate, Houston J. Math., 44(1)(2018), 83-95.
  30. E. Yasar and S. Yalcin, Coefficient inequalities for certain classes of Sakaguchi-type harmonic functions, Southeast Asian Bull. Math., 38(6)(2014), 925-931.