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Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions

Year 2022, Volume: 51 Issue: 6, 1630 - 1636, 01.12.2022
https://doi.org/10.15672/hujms.1072813

Abstract

In this research, we develop some differential subordination results involving harmonic means of $f_{b}(z),f_{b}(z)+zf_{b}^{\prime}(z)$ and $f_{b}(z)+\frac{zf_{b}^{\prime}(z)}{f_{b}(z)},$ where $f_{b}(z)=\frac{z}{\left(1-z^{n}\right) ^{b}},$ $b\geq0;n\in\mathbb{N}=1,2,3...$ is an $n$-fold symmetric Koebe type functions defined in the unit disk with $f_{b}(0)=0,f_{b}^{\prime}(z)\neq0.$ By using the admissibility conditions, we also study several applications in the geometric function theory.

References

  • [1] R.M. Ali, S. Nagpal and V. Ravichandran, Second order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc. 4, 611-629, 2011.
  • [2] I. Graham and D. Varolin, Bloch constans in one and several variables, Pacific J. Math. 174, 347-357, 1996.
  • [3] S. Kanas and J. Stankiewicz, Arithmetic mean with reference to convexity conditions, Bull. Soc. Sci. Lett. Lodz 47, Ser. Rech. Deform. 24, 73-82, 1997.
  • [4] Y.C. Kim and A. Lecko, On differential subordination related to convex functions, J. Math. Anal. Appl. 235, 130-141, 1999.
  • [5] A. Lecko and M. Lecko, Differential subordination of arithmetic and geometric means of some functional related to sector, Int. J. Math. Math. Sci. 2011, Art. ID 205845, 2011.
  • [6] Z. Lewandowski, S.S. Miller and E. Zlotkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56, 111-117, 1976.
  • [7] J.-L. Liu, Certain sufficient conditions for strongly starlike functions associated with an integral operator, Bull. Malays. Math. Sci. Soc. 34, 21-30, 2011.
  • [8] S.S. Miller, and P.T. Mocanu, Differential subordination: Theory and Applications, Dekker, New York, 2000.
  • [9] S.R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malyas. Math. Sci. Soc. 35, 179-194, 2012.
  • [10] R. Omar and S.A. Halim, Multivalent harmonic functions defined by Dziok-Srivastava operator, Bull. Malays. Math. Sci. Soc. 35, 601-610, 2012.
  • [11] S. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc. 49, 109- 115, 1975.
  • [12] K. Stanisława and A.-E. Tudor, Differential subordinations and harmonic means, Bull. Malays. Math. Sci. Soc. 38, 1243–1253, 2015.
Year 2022, Volume: 51 Issue: 6, 1630 - 1636, 01.12.2022
https://doi.org/10.15672/hujms.1072813

Abstract

References

  • [1] R.M. Ali, S. Nagpal and V. Ravichandran, Second order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc. 4, 611-629, 2011.
  • [2] I. Graham and D. Varolin, Bloch constans in one and several variables, Pacific J. Math. 174, 347-357, 1996.
  • [3] S. Kanas and J. Stankiewicz, Arithmetic mean with reference to convexity conditions, Bull. Soc. Sci. Lett. Lodz 47, Ser. Rech. Deform. 24, 73-82, 1997.
  • [4] Y.C. Kim and A. Lecko, On differential subordination related to convex functions, J. Math. Anal. Appl. 235, 130-141, 1999.
  • [5] A. Lecko and M. Lecko, Differential subordination of arithmetic and geometric means of some functional related to sector, Int. J. Math. Math. Sci. 2011, Art. ID 205845, 2011.
  • [6] Z. Lewandowski, S.S. Miller and E. Zlotkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56, 111-117, 1976.
  • [7] J.-L. Liu, Certain sufficient conditions for strongly starlike functions associated with an integral operator, Bull. Malays. Math. Sci. Soc. 34, 21-30, 2011.
  • [8] S.S. Miller, and P.T. Mocanu, Differential subordination: Theory and Applications, Dekker, New York, 2000.
  • [9] S.R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malyas. Math. Sci. Soc. 35, 179-194, 2012.
  • [10] R. Omar and S.A. Halim, Multivalent harmonic functions defined by Dziok-Srivastava operator, Bull. Malays. Math. Sci. Soc. 35, 601-610, 2012.
  • [11] S. Ruscheweyh, New criteria for univalent functions, Proc. Am. Math. Soc. 49, 109- 115, 1975.
  • [12] K. Stanisława and A.-E. Tudor, Differential subordinations and harmonic means, Bull. Malays. Math. Sci. Soc. 38, 1243–1253, 2015.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Syed Zakar Hussain Bukhari 0000-0002-5243-4252

Maryam Nazir This is me 0000-0001-8382-0204

Publication Date December 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 6

Cite

APA Hussain Bukhari, S. Z., & Nazir, M. (2022). Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions. Hacettepe Journal of Mathematics and Statistics, 51(6), 1630-1636. https://doi.org/10.15672/hujms.1072813
AMA Hussain Bukhari SZ, Nazir M. Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions. Hacettepe Journal of Mathematics and Statistics. December 2022;51(6):1630-1636. doi:10.15672/hujms.1072813
Chicago Hussain Bukhari, Syed Zakar, and Maryam Nazir. “Differential Subordination and Harmonic Means for the Koebe Type $n$-Fold Symmetric Functions”. Hacettepe Journal of Mathematics and Statistics 51, no. 6 (December 2022): 1630-36. https://doi.org/10.15672/hujms.1072813.
EndNote Hussain Bukhari SZ, Nazir M (December 1, 2022) Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions. Hacettepe Journal of Mathematics and Statistics 51 6 1630–1636.
IEEE S. Z. Hussain Bukhari and M. Nazir, “Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 6, pp. 1630–1636, 2022, doi: 10.15672/hujms.1072813.
ISNAD Hussain Bukhari, Syed Zakar - Nazir, Maryam. “Differential Subordination and Harmonic Means for the Koebe Type $n$-Fold Symmetric Functions”. Hacettepe Journal of Mathematics and Statistics 51/6 (December 2022), 1630-1636. https://doi.org/10.15672/hujms.1072813.
JAMA Hussain Bukhari SZ, Nazir M. Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions. Hacettepe Journal of Mathematics and Statistics. 2022;51:1630–1636.
MLA Hussain Bukhari, Syed Zakar and Maryam Nazir. “Differential Subordination and Harmonic Means for the Koebe Type $n$-Fold Symmetric Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 6, 2022, pp. 1630-6, doi:10.15672/hujms.1072813.
Vancouver Hussain Bukhari SZ, Nazir M. Differential subordination and harmonic means for the Koebe type $n$-fold symmetric functions. Hacettepe Journal of Mathematics and Statistics. 2022;51(6):1630-6.