Skip to main content
SpringerLink
Log in

Some Inequalities and Other Results Associated with Certain Subclasses of Univalent and Bi-Univalent Analytic Functions

  • Chapter
Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 68))

Abstract

In recent year, various interesting properties and characteristics (including, for example, coefficient bounds and coefficient inequalities) of many different subclasses of univalent and bi-univalent analytic functions have been systematically investigated. The main object of this essentially survey-cum-expository article is first to present a brief account of some important contributions to the theory of univalent and bi-univalent analytic functions, which have been made in several recent works. References to other more recent investigations involving many closely-related function classes are also provided for motivating and encouraging future researches on these topics in Geometric Function Theory of Complex Analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alexander, J.W.: Functions which map the interior of the unit circle upon simple regions. Ann. Math. (Ser. 2) 17, 12–22 (1915)

    MATH  Google Scholar 

  2. Altıntaş, O., Irmak, H., Owa, S., Srivastava, H.M.: Coefficient bounds for some families of starlike and convex functions of complex order. Appl. Math. Lett. 20, 1218–1222 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Altintaş, O., Irmak, H., Srivastava, H.M.: Fractional calculus and certain starlike functions with negative coefficients. Comput. Math. Appl. 30(2), 9–15 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. Altintaş, O., Özkan, Ö.: Starlike, convex and close-to-convex functions of complex order. Hacettepe Bull. Nat. Sci. Eng. Ser. B 28, 37–46 (1991)

    Google Scholar 

  5. Altintaş, O., Özkan, Ö.: On the classes of starlike and convex functions of complex order. Hacettepe Bull. Nat. Sci. Eng. Ser. B 30, 63–68 (2001)

    MATH  Google Scholar 

  6. Altintaş, O., Özkan, Ö., Srivastava, H.M.: Neighborhoods of a class of analytic functions with negative coefficients. Appl. Math. Lett. 13(3), 63–67 (1995)

    Article  Google Scholar 

  7. Altintaş, O., Özkan, Ö., Srivastava, H.M.: Majorization by starlike functions of complex order. Complex Var. Theory Appl. 46, 207–218 (2001)

    Article  MATH  Google Scholar 

  8. Altintaş, O., Özkan, Ö., Srivastava, H.M.: Neighborhoods of a certain family of multivalent functions with negative coefficient. Comput. Math. Appl. 47, 1667–1672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Altintaş, O., Srivastava, H.M.: Some majorization problems associated with p-valently starlike and convex functions of complex order. East Asian Math. J. 17, 175–183 (2001)

    MATH  Google Scholar 

  10. Brannan, D.A., Clunie, J.G. (eds.): Aspects of Contemporary Complex Analysis. Proceedings of the NATO Advanced Study Institute (University of Durham, Durham; July 1–20, 1979). Academic Press, New York (1980)

    MATH  Google Scholar 

  11. Brannan, D.A., Clunie, J., Kirwan, W.E.: Coefficient estimates for a class of star-like functions. Can. J. Math. 22, 476–485 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brannan, D.A., Taha, T.S.: On some classes of bi-unvalent functions. In: Mazhar, S.M., Hamoui, A., Faour, N.S. (eds.) Mathematical Analysis and Its Applications, Kuwait, February 18–21, 1985. KFAS Proceedings Series, vol. 3, pp. 53–60. Pergamon, Oxford (1988); see also Studia Univ. Babeş-Bolyai Math. 31(2), 70–77 (1986)

    Google Scholar 

  13. Breaz, D., Breaz, N.: The univalent conditions for an integral operator on the classes and . J. Approx. Theory Appl. 1, 93–98 (2005)

    MathSciNet  Google Scholar 

  14. Breaz, D., Breaz, N.: Univalence of an integral operator. Mathematica (Cluj) 47(70), 35–38 (2005)

    MathSciNet  Google Scholar 

  15. Breaz, D., Breaz, N., Srivastava, H.M.: An extension of the univalent condition for a family of integral operators. Appl. Math. Lett. 22, 41–44 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deng, Q.: Certain subclass of analytic functions with complex order. Appl. Math. Comput. 208, 359–362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duren, P.L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)

    MATH  Google Scholar 

  18. Gao, C., Zhou, S.: On a class of analytic functions related to the starlike functions. Kyungpook Math. J. 45, 123–130 (2005)

    MathSciNet  MATH  Google Scholar 

  19. Kowalczyk, J., Leś-Bomba, E.: On a subclass of close-to-convex functions. Appl. Math. Lett. 23(10), 1147–1151 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Miller, S.S., Mocanu, P.T.: Differential Subordination: Theory and Applications. Series on Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Dekker, New York (2000)

    MATH  Google Scholar 

  22. Milovanović, G.V., Mitrinović, D.S., Rassias, Th.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  23. Moldoveanu, S., Pascu, N.N.: Integral operators which preserve the univalence. Mathematica (Cluj) 32(55), 159–166 (1990)

    MathSciNet  Google Scholar 

  24. Nasr, M.A., Aouf, M.K.: Radius of convexity for the class of starlike functions of complex order. Bull. Fac. Sci. Assiut Univ., Sect. a Nat. Sci. 12, 153–159 (1983)

    MathSciNet  Google Scholar 

  25. Nehari, Z.: Conformal Mapping. McGraw-Hill, New York (1952). Reprinted by Dover Publications Incorporated, New York (1975)

    MATH  Google Scholar 

  26. Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|<1. Arch. Ration. Mech. Anal. 32, 100–112 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  27. Owa, S., Nunokawa, M., Saitoh, H., Srivastava, H.M.: Close-to-convexity, starlikeness, and convexity of certain analytic functions. Appl. Math. Lett. 15, 63–69 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ozaki, S., Nunokawa, M.: The Schwarzian derivative and univalent functions. Proc. Am. Math. Soc. 33, 392–394 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pascu, N.N.: On a univalence criterion. II. In: Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1985), pp. 153–154. Preprint 86–6, University “Babeş-Bolyai”, Cluj-Napoca, 1985

    Google Scholar 

  30. Pescar, V.: New criteria for univalence of certain integral operators. Demonstr. Math. 33, 51–54 (2000)

    MathSciNet  MATH  Google Scholar 

  31. Pescar, V.: On the univalence of an integral operator. Appl. Math. Lett. 23, 615–619 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pescar, V., Breaz, D.: Some integral operators and their univalence. Acta Univ. Apulensis, Mat.-Inform. 15, 147–152 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Pescar, V., Breaz, D.: On an integral operator. Appl. Math. Lett. 23, 625–629 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rassias, Th.M. (ed.): Constantin Carathéodory (1873–1950): An International Tribute, vols. I and II. World Scientific, Singapore (1991)

    Google Scholar 

  35. Rassias, Th.M., Srivastava, H.M. (eds.): Analytic and Geometric Inequalities and Applications. Series on Mathematics and Its Applications, vol. 478. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  36. Robertson, M.S.: On the theory of univalent functions. Ann. Math. (Ser. 1) 37, 374–408 (1936)

    Google Scholar 

  37. Rogosinski, W.: On the coefficients of subordinate functions. Proc. Lond. Math. Soc. (Ser. 2) 48, 48–82 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  38. Sălăgean, G.Ş.: Subclass of univalent functions. In: Complex Analysis: Fifth Romanian–Finnish Seminar, Part 1, Bucharest, 1981. Lecture Notes in Mathematics, vol. 1013, pp. 362–372. Springer, Berlin (1983)

    Chapter  Google Scholar 

  39. Srivastava, H.M., Altıntaş, O., Kırcı Serenbay, S.: Coefficient bounds for certain subclasses of starlike functions of complex order. Appl. Math. Lett. 24, 1359–1363 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Srivastava, H.M., Deniz, E., Orhan, H.: Some general univalence criteria for a family of integral operators. Appl. Math. Comput. 215, 3696–3701 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Srivastava, H.M., Eker, S.S.: Some applications of a subordination theorem for a class of analytic functions. Appl. Math. Lett. 21, 394–399 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Srivastava, H.M., Eker, S.S., Şeker, B.: A certain convolution approach for subclasses of analytic functions with negative coefficients. Integral Transforms Spec. Funct. 20, 687–699 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010). doi:10.1016/j.aml.2010.05.009

    Article  MathSciNet  MATH  Google Scholar 

  44. Srivastava, H.M., Owa, S. (eds.): Univalent Functions, Fractional Calculus, and Their Applications. Halsted, New York (1989)

    MATH  Google Scholar 

  45. Srivastava, H.M., Owa, S. (eds.): Current Topics in Analytic Function Theory. World Scientific, Singapore (1992)

    MATH  Google Scholar 

  46. Srivastava, H.M., Xu, Q.-H., Wu, G.-P.: Coefficient estimates for certain subclasses of spiral-like functions of complex order. Appl. Math. Lett. 23, 763–768 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Srivastava, H.M., Yang, D.-G., Xu, N.-E.: Subordinations for multivalent analytic functions associated with the Dziok-Srivastava operator. Integral Transforms Spec. Funct. 20, 581–606 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Taha, T.S.: Topics in univalent function theory. Ph.D. Thesis. University of London (1981)

    Google Scholar 

  49. Xu, Q.-H., Gui, Y.-C., Srivastava, H.M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25, 990–994 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  50. Xu, Q.-H., Gui, Y.-C., Srivastava, H.M.: Coefficient estimates for certain subclasses of analytic functions of complex order. Taiwan. J. Math. 15, 2377–2386 (2011)

    MathSciNet  MATH  Google Scholar 

  51. Xu, Q.-H., Srivastava, H.M., Li, Z.: A certain subclass of analytic and close-to-convex functions. Appl. Math. Lett. 24, 396–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

It is a great pleasure for me to dedicate this article to Prof. Dr. Themistocles Michael Rassias on the occasion of his 60th birthday. The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada

    H. M. Srivastava

Authors
  1. H. M. Srivastava
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. M. Srivastava .

Editor information

Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Department of Mathematics & Statistics, University of Victoria, Finnerty Road 3800, Victoria, V8W 3R4, British Columbia, Canada

    Hari M. Srivastava

Additional information

Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Srivastava, H.M. (2012). Some Inequalities and Other Results Associated with Certain Subclasses of Univalent and Bi-Univalent Analytic Functions. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_38

Download citation

Publish with us

Policies and ethics

Search

Navigation