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Pre-Schwarzian norm estimates for the class of janowski starlike functions

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Abstract

For \(-1\le B<A\le 1\), let \({\mathcal {S}}^*(A,B)\) denote the class of Janowski starlike functions that satisfy the subordination relation \(zf'(z)/f(z)\prec (1+Az)/(1+Bz)\). In the present article, we determine the sharp estimate of the pre-Schwarzian norm for the functions in the class \({\mathcal {S}}^*(A,B)\).

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References

  1. Ali, M.F., Vasudevarao, A.: On coefficient estimates of negative powers and inverse coefficients for certain starlike functions. Proc. Indian Acad. Sci. (Math. Sci.) 127(3), 449–462 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aghalary, R., Orouji, Z.: Norm Estimates of the Pre-Schwarzian Derivatives for \(\alpha \)-Spiral-Like Functions of Order \(\rho \). Complex Anal. Oper. Theory 8(4), 791–801 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker, J.: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. Reine Angew. Math. 255, 23–43 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Becker, J., Pommerenke, C.: Schlichtheitskriterien und Jordangebiete. J. Reine Angew. Math. 354, 74–94 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Chuaqui, M., Duren, P., Osgood, B.: The Schwarzian derivative for harmonic mappings. J. Anal. Math. 91(1), 329–351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chuaqui, M., Hernández, R., Martín, M.J.: Affine and linear invariant families of harmonic mappings. Math. Ann. 367(3), 1099–1122 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Choi, J.H., Kim, Y.C., Ponnusamy, S., Sugawa, T.: Norm estimates for the Alexander transforms of convex functions of order alpha. J. Math. Anal. Appl. 303, 661–668 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duren, P.L.: Univalent functions (Grundlehren der mathematischen Wissenschaften 259. Berlin, Heidelberg, Tokyo), Springer-Verlag, New York (1983)

  9. Goodman, A.W.: Univalent Functions, Vols. I and II. Mariner Publishing Co. Tampa, Florida (1983)

  10. Hernández, R., Martín, M.J.: Pre-Schwarzian and Schwarzian derivatives of harmonic mappings. J. Geom. Anal. 25(1), 64–91 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Janowski, W.: Some extremal problems for certain families of analytic functions. Ann. Polon. Math. 28, 297–326 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim, Y.C., Sugawa, T.: Growth and coefficient estimates for uniformly locally univalent functions on the unit disk. Rocky Mountain J. Math. 32, 179–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, Y.C., Sugawa, T.: Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions. Proc. Edinb. Math. Soc. (2) 49(1), 131–143 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kraus, W.: Über den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung. Mitt. Math. Sem. Giessen 21, 1–28 (1932)

    MATH  Google Scholar 

  16. Liu, G., Ponnusamy, S.: Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm. Indag. Math. (N.S.) 29(2), 752–778 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nehari, Z.: The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc. 55(6), 545–551 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  18. Okuyama, Y.: The norm estimates of pre-Schwarzian derivatives of spiral-like functions. Complex Var. Theory Appl. 42, 225–239 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Padmanabhan, K.S.: On certain classes of starlike functions in the unit disk. J. Indian Math. Soc., (N.S.) 32, 89–103 (1968)

    MathSciNet  MATH  Google Scholar 

  20. Parvatham, R., Ponnusamy, S., Sahoo, S.K.: Norm estimates for the Bernardi integral transforms of functions defined by subordination. Hiroshima Math. J. 38, 19–29 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ponnusamy, S., Juneja, O.P.: Some applications of Briot-Bouquet differential subordinations. Ann. Univ. Mariae Curie-Skłodowska Sect. A. 42, 129–144 (1988)

    MathSciNet  MATH  Google Scholar 

  22. Ponnusamy, S., Sahoo, S.K.: Norm estimates for convolution transforms of certain classes of analytic functions. J. Math. Anal. Appl. 342, 171–180 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ponnusamy, S., Sahoo, S.K.: Pre-Schwarzian norm estimates of functions for a subclass of strongly starlike functions. Mathematica 52(75), 47–53 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Ponnusamy, S., Sharma, N.L., Wirths, K.J.: Logarithmic coefficients of the inverse of univalent functions. Results Math. 73, 1–20 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ponnusamy, S., Sharma, N.L., Wirths, K.J.: Logarithmic coefficients problems in families related to starlike and convex functions. J. Aust. Math. Soc. 109, 230–249 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ponnusamy, S., Wirths, K.J.: On the problem of Gromova and Vasil’ev on integral means, and Yamashitas conjecture for spirallike functions. Ann. Acad. Sci. Fenn. Math. 39(2), 721–731 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Silverman, H., Silvia, E.M.: Subclasses of starlike functions subordinate to convex functions. Canad. J. Math. 37(1), 48–61 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Singh, R., Singh, V.: On a class of bounded starlike functions. Indian J. Pure Appl. Math. 5, 733–754 (1974)

    MathSciNet  MATH  Google Scholar 

  29. Sugawa, T.: On the norm of pre-Schwarzian derivatives of strongly starlike functions. Ann. Univ. Mariae Curie-Skłodowska Sect. A 52(2), 149–157 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Yamashita, S.: Almost locally univalent functions. Monatsh. Math. 81, 235–240 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yamashita, S.: Norm estimates for function starlike or convex of order alpha. Hokkaido Math. J. 28(1), 217–230 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the referee for the constructive comments which helped to improve the presentation of the paper. The second named author thanks the University Grants Commission for the financial support through UGC Fellowship (Grant No. MAY2018-429303).

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Durgapur, Durgapur, 713209, West Bengal, India

    Md Firoz Ali & Sanjit Pal

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  1. Md Firoz Ali
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  2. Sanjit Pal
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All authors contributed equally to the investigation of the problem and the order of the authors is given alphabetically according to the surname. All authors read and approved the final manuscript.

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Correspondence to Md Firoz Ali.

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Communicated by Adrian Constantin.

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Ali, M.F., Pal, S. Pre-Schwarzian norm estimates for the class of janowski starlike functions. Monatsh Math 201, 311–327 (2023). https://doi.org/10.1007/s00605-022-01756-4

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