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

Year 2022, , 1630 - 1636, 01.12.2022

Syed Zakar Hussain Bukhari , Maryam Nazir

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.

Keywords

Differential subordination, harmonic means, Koebe type n-fold symmetric function

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.