A new subclass of analytic functions

Abstract

In the present paper, we introduce a class $\mathcal{B}_\theta(\alpha,\beta)$ of functions, analytic in $|z|<1$, such that $f(0)=0$, $f'(0)=1$ and
\[\alpha< Re\left(f'(z)+\frac{1+e^{i\theta}}{2}zf''(z)\right)<\beta\quad (|z|<1),\]
where $\theta\in(-\pi,\pi]$, $0\leq\alpha<1$ and $\beta>1$. Integral representation, differential subordination results and coefficient estimates are considered. Also Fekete-Szegö coefficient functional associated with the $k$--th root transform $[f(z^k)]^{1/k}$ for functions in the class $\mathcal{B}_\theta(\alpha,\beta)$ is investigated.

Keywords

• analytic functions
• univalent functions
• Carathéodory functions
• differential subordination
• Fekete-Szegö inequality

References

1. [1] R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramanian, The Fekete-Szegö coefficient functional for transforms of analytic functions, Bull. Iranian Math. Soc. 35, 119–142, 2009.
2. [2] P.N. Chichra, New subclasses of the class of close-to-convex functions, Proc. Amer. Math. Soc. 62, 37–43, 1977.
3. [3] D.J. Hallenbeck and St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc. 52, 191–195, 1975.
4. [4] R. Kargar, A. Ebadian, J. Sokół, On subordination of some analytic functions, Sib. Math. J. 57, 599–605, 2016.
5. [5] R. Kargar, A. Ebadian and J. Sokół, Radius problems for some subclasses of analytic functions, Complex Anal. Oper. Theory 11, 1639–1649, 2017.
6. [6] R. Kargar, A. Ebadian and J. Sokół, On Booth lemniscate and starlike functions, Anal. Math. Phys. 9, 143–154, 2019.
7. [7] F.R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20, 8–12, 1969.
8. [8] K. Kuroki and S. Owa, Notes on New Class for Certain Analytic Functions, RIMS Kokyuroku Kyoto Univ. 1772, 21–25, 2011.

9. [9] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49, 109–115, 1975.
10. [10] St. Ruscheweyh and T. Sheil-Small, Hadamard products of schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv. 48, 119–135, 1973.
11. [11] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48, 48–82, 1943.
12. [12] H. Silverman, A class of bounded starlike functions, Internat. J. Math. Math. Sci. 17, 249–252, 1994.
13. [13] H. Silverman and E.M. Silvia, Characterizations for subclasses of univalent functions, Math. Japan. 50, 103–109, 1999.
14. [14] R. Singh and S. Singh, Convolutions properties of a class of starlike functions, Proc. Amer. Math. Soc. 106, 145–152, 1989.