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Starlikeness and Convexity of Certain Integral Transforms by using Duality Technique

  • Satwanti Devi
  • A. Swaminathan
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Duality technique is applied on the integral transform of the type \(V_\lambda(f)(z)=\int_0^1\lambda(t)\frac{f(tz)}{t}{} dt\), where \(\int_0^1 \lambda(t)dt =1\) and f is functional class that satisfies certain analytic characterization in the unit disk. This leads to the investigation between the equivalence of non-negativity of a linear functional and the given integral transform involving starlike and convex functions. Particular values of \(\lambda(t)\) give rise to well-known integral operators. Investigation of the parameters for such values leads to interesting results in univalent function theory. This chapter outlines all the possible results available in the literature in this direction to provide the reader an overview.

Keywords

Integral Operator Open Unit Disk Application Purpose Starlike Function Important Subclass 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

The authors wish to thank the anonymous referee for the valuable comments and suggestions to improve the quality of the chapter.

References

  1. 1.
    Alexander, J.W.: Functions which map the interior of the unit circle upon simple regions. Ann. Math. (2) 17(1), 12–22 (1915)CrossRefMATHGoogle Scholar
  2. 2.
    Ali, R.M., Singh, V.: Convexity and starlikeness of functions defined by a class of integral operators. Complex Var. Theory Appl. 26(4), 299–309 (1995)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Ali, R.M., Badghaish, A.O., Ravichandran, V., Swaminathan, A.: Starlikeness of integral transforms and duality. J. Math. Anal. Appl. 385(2), 808–822 (2012)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Ali, R.M., Nargesi, M.M., Ravichandran, V.: Convexity of integral transforms and duality. Complex Var. Ellip. Equ. 58(11), 1–22 (2012). doi:10.1080/17476933.2012.693483MathSciNetGoogle Scholar
  5. 5.
    Balasubramanian, R., Ponnusamy, S., Prabhakaran, D.J.: Duality techniques for certain integral transforms to be starlike. J. Math. Anal. Appl. 293(1), 355–373 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Balasubramanian, R., Ponnusamy, S., Prabhakaran, D.J.: Convexity of integral transforms and function spaces. Integr. Transform. Spec. Funct. 18(1–2), 1–14 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Balasubramanian, R., Ponnusamy, S., Prabhakaran, D.J.: On extremal problems related to integral transforms of a class of analytic functions. J. Math. Anal. Appl. 336(1), 542–555 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Barnard, R.W., Naik, S., Ponnusamy, S.: Univalency of weighted integral transforms of certain functions. J. Comput. Appl. Math. 193(2), 638–651 (2006)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bernardi, S.D.: Convex and starlike univalent functions. Trans. Am. Math. Soc. 135, 429–446 (1969)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Carlson, B.C., Shaffer, D.B.: Starlike and prestarlike hypergeometric functions. SIAM J. Math. Anal. 15(4), 737–745 (1984)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Choi, J.H., Kim, Y.C., Saigo, M.: Geometric properties of convolution operators defined by Gaussian hypergeometric functions. Integr. Transform. Spec. Funct. 13(2), 117–130 (2002)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154(1–2), 137–152 (1985)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Devi S, Swaminathan A (2014) Integral transforms of functions to be in a class of analytic functions using duality techniques. J Complex Anal.10 p. Art. ID 473069Google Scholar
  14. 14.
    Duren, P.L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)MATHGoogle Scholar
  15. 15.
    Fournier, R., Ruscheweyh, S.: On two extremal problems related to univalent functions. Rocky Mt. J. Math. 24(2), 529–538 (1994)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Grunsky, H.: Neue Abschtzungen zur konformen Abbildung ein-und mehrfach zusammenhngender Bereiche. Schr. Math. Inst. Inst. Angew. Math. Univ. Berl 1, 95–140 (1932)Google Scholar
  17. 17.
    Hohlov, Y.E.: Convolution operators preserving univalent functions. Pliska Stud. Math. Bulg. 10, 87–92 (1989)MATHMathSciNetGoogle Scholar
  18. 18.
    Kaplan, W.: Close-to-convex schlicht functions. Mich. Math. J. 1(1952), 169–185 (1953)Google Scholar
  19. 19.
    Kim, Y.C., Rønning, F.: Integral transforms of certain subclasses of analytic functions. J. Math. Anal. Appl. 258(2), 466–489 (2001)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Komatu, Y.: On analytic prolongation of a family of operators. Mathematica (Cluj) (2)32(55), 141–145 (1990)MathSciNetGoogle Scholar
  21. 21.
    Libera, R.J.: Some classes of regular univalent functions. Proc. Am. Math. Soc. 16(4), 755–758 (1965)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Nunokawa, M., Thomas, D.K.: On the Bernardi integral operator. Current Topics in Analytic Function Theory, pp. 212–219. World Scientific, River Edge (1992)CrossRefGoogle Scholar
  23. 23.
    Omar, R., Halim, S.A., Ibrahim, R.W.: Starlikeness of Order for Certain Integral Transforms Using Duality (Pre-print)Google Scholar
  24. 24.
    Omar, R., Halim, S.A., Ibrahim, R.W.: Convexity and Starlikeness of Certain Integral Transforms Using Duality (Pre-print)Google Scholar
  25. 25.
    Pascu, N.N., Podaru, V.: On the radius of alpha-starlikeness for starlike functions of order beta. Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013. 336–349. Springer, Berlin (1983)Google Scholar
  26. 26.
    Ponnusamy, S.: Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane. Rocky Mt. J. Math. 28(2), 695–733 (1998)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Ponnusamy, S., Rønning, F.: Duality for Hadamard products applied to certain integral transforms. Complex Var. Theory Appl. 32(3), 263–287 (1997)CrossRefMATHGoogle Scholar
  28. 28.
    Ponnusamy, S., Rønning, F.: Integral transforms of a class of analytic functions. Complex Var. Ellip. Equ. 53(5), 423–434 (2008)CrossRefMATHGoogle Scholar
  29. 29.
    Raghavendar, K., Swaminathan, A.: Integral transforms of functions to be in certain class defined by the combination of starlike and convex functions. Comput. Math. Appl. 63(8), 1296–1304 (2012)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Verma, S., Gupta, S., Singh, S.: On an integral transform of a class of analytic functions. Abstr. Appl. Anal. 2012, 10 (2012). (Art. ID 259054, MR2984036)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Verma, S., Gupta, S., Singh, S.: Order of convexity of integral transforms and duality (May 2013). (arXiv:1305.0732)Google Scholar
  32. 32.
    Verma, S., Gupta, S., Singh, S.: Duality and integral transform of a class of analytic functions. Bull. Malays. Math. Soc. (To appear)Google Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyUttarakhandIndia

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