Hadamard product of holomorphic mappings associated with the conic shaped domain

Abstract

We define certain subclasses $\delta-\mathcal{UM}(\ell,\eta_{1},\eta_{2})$ and $\delta-\mathcal{UM}_{\Im}(\ell,\eta_{1},\eta_{2})$ of holomorphic mappings involving some differential inequalities. These functions are actually generalizations of some basic families of starlike and convex mappings. We study sufficient conditions for $f\in \delta-\mathcal{UM}(\ell,\eta_{1}% ,\eta_{2}).$ We also discuss the characterization for $f\in \delta -\mathcal{UM}_{\Im}(\ell,\eta_{1},\eta_{2})$ along with the coefficient bounds and other problems. Using certain conditions for functions in the class $\delta-\mathcal{UM}(\ell,\eta_{1},\eta_{2}),$ we also define another class $\delta-\mathcal{UM}^{\ast}(\ell,\eta_{1},\eta_{2})$ and study some subordination related result.

Keywords

• Caratheodory functions
• convolution
• subordination

References

1. Aouf, M. K., Mostafa, A. O., Some properties of a subclass of uniformly convex functions with negative coefficients, Demonstration Math., 2 (2008), 353-370. https://doi.org/10.1515/dema-2008-0212
2. Aouf, M. K., El-Ashwah, R. M., El-Deeb, S. M., Subordination results for certain subclasses of uniformly starlike and convex functions defined by convolution, European J. Pure Appl. Math., 3(5) (2010), 903-917.
3. Attiya, A. A., On some application of a subordination theorems, J. Math. Anal. Appl., 311 (2005), 489-494. https://doi.org/10.1016/j.jmaa.2005.02.056
4. Bukhari, S. Z. H., Bulboaca, T., Shabbir, M. S., Subordination and superordination results for analytic functions with respect to symmetrical points, Quaest. Math., 41(1) (2018), 1-15. https://doi.org/10.2989/16073606.2017.1372528
5. Bukhari, S. Z. H., Raza, M., Nazir, M., Some generalizations of the class of analytic functions with respect to δ-symmetric points, Hacet. J. Math. Stat., 45(1) (2016), 1-14.
6. Bukhari, S. Z. H., Sokol, J., Zafar, S., Unified approach to starlike and convex functions involving convolution between analytic functions, Results in Math., 73 (2018), 30. https://doi.org/10.1007/s00025-018-0782-0
7. Goodman, A. W., On uniformly starlike functions, J. Math. Anal. Appl., 155 (1991), 364-370. https://doi.org/10.1016/0022-247X(91)90006-L
8. Janowski, W., Some extremal problems for certain classes of analytic functions, Ann. Polon. Math., 28 (1973), 297-326. 10.4064/AP-28-3-297-326

9. Kanas, S., Wisniowska, A., Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl., 45 (2000), 647-657.
10. Miller, S. S., Mocanu, P. T., Differential Subordinations Theory and Applications, Series on Monographs and Textbooks in Pure and Appl Math, No. 255 Marcel Dekker, Inc., New York, 2000. https://doi.org/10.1201/9781482289817
11. Noor, K. I., Malik, S. N., On coefficient inequalities of functions associated with conic domains, Comp. Math. Appl., 62 (2011), 2209-2217. https://doi.org/10.1016/j.camwa.2011.07.006
12. Raina, R. K., Deepak, B., Some properties of a new class of analytic functions defined in terms of a Hadamard product, J. Inequal Pure Appl. Math., 9 (2008), 1-9.
13. Ronning, F., Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189-196. https://doi.org/10.2307/2160026
14. Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109-116.
15. Srivastava, H. M., Attiya, A. A., Some subordination results associated with certain subclass of analytic functions, J. Inequal Pure Appl. Math., 5(4) (2004), 1-6.
16. Wilf, H. S., Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 129 (1961), 689-693.