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Preserving properties and pre-Schwarzian norms of nonlinear integral transforms

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Abstract

We study preserving properties of certain nonlinear integral transforms in some classical families of normalized analytic univalent functions defined in the unit disk. Also, we find sharp pre-Schwarzian norm estimates of such integrals.

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References

  1. 1.

    Aksent’ev, L.A., Nezhmetdinov, I.R., Sufficient conditions for univalence of certain integral transforms, Tr. Semin. Kraev. Zadacham. Kazan, 18, : 3–11 (in Russian); translation. Amer. Math. Soc. Transl. 136 (1987), 1–9 (1982)

  2. 2.

    Md Firoz Ali and A. Vasudevarao, On certain families of analytic functions in the Hornich space, Comput. Methods Funct. Theory, 18 (2018), 643–659

  3. 3.

    P. L. Duren, Univalent Functions, Springer-Verlag (New York, 1983)

  4. 4.

    Goodman, A.W.: Univalent Functions, 1, 2, Mariner Publishing Co. Tampa, FL (1983)

  5. 5.

    Hartmann, F.W., MacGregor, T.H.: Matrix transformations of univalent power series. J. Aust. Math. Soc. 18, 419–435 (1974)

  6. 6.

    Kim, Y.J., Merkes, E.P.: On an integral of powers of a spirallike function. Kyungpook Math. J. 12, 249–253 (1972)

  7. 7.

    Kim, Y.C., Ponnusamy, S., Sugawa, T.: Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives. J. Math. Anal. Appl. 299, 433–447 (2004)

  8. 8.

    Y. C. Kim, S. Ponnusamy and T. Sugawa, Geometric properties of nonlinear integral transforms of certain analytic functions, Proc. Japan Acad., 80, Ser. A (2004), 57–60

  9. 9.

    Kim, Y.C., Srivastava, H.M.: Geometric properties of certain non-linear integral operators. Int. Transform Spec. Funct. 17, 723–732 (2006)

  10. 10.

    Kim, Y.C., Sugawa, T.: The Alexander transform of a spirallike function. J. Math. Anal. Appl. 325, 608–611 (2007)

  11. 11.

    Koepf, W.: Classical families of univalent functions in the Hornich space. Monatsh. Math. 100, 113–120 (1985)

  12. 12.

    Shankey Kumar and S. K. Sahoo, Properties of \(\beta \)-Cesàro operators on \(\alpha \)-Bloch space, arxiv:1808.08844

  13. 13.

    Libera, R.J.: Univalent \(\alpha \)-spiral functions. Canad. J. Math. 19, 449–456 (1967)

  14. 14.

    Li, L., Ponnusamy, S., Qiao, J.: Generalized Zalcman conjecture for convex functions of order \(\alpha \). Acta Math. Hungar. 150, 234–246 (2016)

  15. 15.

    Merkes, E.P.: Univalence of an integral transform. Contemp. Math. 38, 113–119 (1985)

  16. 16.

    Merkes, E.P., Wright, D.J.: On the univalence of a certain integral. Proc. Amer. Math. Soc. 27, 97–100 (1971)

  17. 17.

    S. S. Miller and P. T. Mocanu, Differential Subordinations – Theory and Applications, Marcel Dekker, Inc. (New York, 2000)

  18. 18.

    Nunokawa, M.: On the univalence of a certain integral. Proc. Japan Acad. 45, 841–845 (1969)

  19. 19.

    Pfaltzgraff, J.A.: Univalence of the integral of \(f^{\prime }(z)^\lambda \). Bull. London Math. Soc. 7, 254–256 (1975)

  20. 20.

    J. A. Pfaltzgraff, M. O. Reade and T. Umezawa, Sufficient conditions for univalence, Ann. Fac. Si. de Kinshasa, Zaïre, Sect. Math. Phys., 2 (1976), 211–218

  21. 21.

    Ponnusamy, S., Sahoo, S.K., Sugawa, T.: Hornich operations on functions of bounded boundary rotations and order \(\alpha\). Comput. Methods Funct. Theory 19, 455–472 (2019)

  22. 22.

    Ponnusamy, S., Wirths, K.-J.: On the problem of Gromova and Vasil'ev on integral means, and Yamashita's conjecture for spirallike functions. Ann. Acad. Sci. Fenn. Math. 39, 721–731 (2014)

  23. 23.

    Royster, W.C.: On the univalence of a certain integral. Michigan Math. J. 12, 385–387 (1965)

  24. 24.

    Singh, V., Chichra, P.N.: An extension of Becker's criterion of univalence. J. Indian Math. Soc. 41, 353–361 (1977)

  25. 25.

    Umezawa, T.: Analytic functions convex in one direction. J. Math. Soc. Japan 4, 194–202 (1952)

  26. 26.

    Yamashita, S.: Norm estimates for function starlike or convex of order alpha. Hokkaido Math. J. 28, 217–230 (1999)

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Acknowledgement

The authors thank Professor Toshiyuki Sugawa for his useful remarks leading to some improvement in the paper.

Author information

Correspondence to S. K. Sahoo.

Additional information

The work of the first author is supported by CSIR, New Delhi (Grant No. 09/1022(0034)/2017-EMR-I).

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Kumar, S., Sahoo, S.K. Preserving properties and pre-Schwarzian norms of nonlinear integral transforms. Acta Math. Hungar. (2020). https://doi.org/10.1007/s10474-020-01027-4

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Key words and phrases

  • integral transform
  • Hornich operator
  • Cesàro transform
  • pre-Schwarzian norm
  • univalent function
  • spirallike function
  • convex function
  • close-to-convex function

Mathematics Subject Classification

  • primary 30C55
  • 35A22
  • secondary 30C45
  • 35A23
  • 65R10