New subclass of analytic functions defined by subordination

Document Type : Research Paper


Department of Mathematics, Payame Noor University, Tehran, Iran


By using the subordination relation $"\prec"$, we introduce an interesting subclass of analytic functions as follows:
\mathcal{S}^*_{\alpha}:=\left\{f\in \mathcal{A}:\frac{zf'(z)}{f(z)}\prec \frac{1}{(1-z)^\alpha},\ \ |z|<1\right\},
where $0<\alpha\leq1$ and $\mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. In the present paper, by the class $\mathcal{S}^*_{\alpha}$ and by the Nunokawa lemma we generalize a famous result connected to starlike functions of order $1/2$. Also, coefficients inequality and logarithmic coefficients inequality for functions of the class $\mathcal{S}^*_{\alpha}$ are obtained.


[1] L. De Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985) 137–152.
[2] O. Dvorak, On funkcich prosty, Casopis Pest. Mat. 63 (1934) 9–16.
[3] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1971) 159–177.
[4] R. Kargar, A. Ebadian and J. Sokol, On Booth lemniscate and starlike functions, Anal. Math. Phys. 9 (2019) 143–154.
[5] R. Kargar, A. Ebadian and J. Sokol, Radius problems for some subclasses of analytic functions, Complex Anal. Oper. Theory 11 (2017) 1639–1649.
[6] S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2016) 199–212.
[7] K. Kuroki and S. Owa, Notes on new class for certain analytic functions, RIMS Kokyuroku Kyoto Univ. 1772 (2011) 2125.
[8] W. Ma and, D. Minda, A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Z. Li, F. Ren, L. Yang, and S. Zhang, eds.), International Press Inc., 1992, pp. 157–169.
[9] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992) 165–175.
[10] R. Mendiratta, S. Nagpal and V. Ravichandran, A subclass of starlike functions associated with left–half of the lemniscate of Bernoulli, Int. J. Math. 25 (2014), 17 pp.
[11] R. Mendiratta, S. Nagpal and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. 38 (2015) 365–386.
[12] K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. 2 (1934–35) 129–155.
[13] M. Nunokawa, On properties of non-Caratheodory functions, Proc. Jpn. Acad., Ser. A, Math. Sci. 68 (1992) 152–153.
[14] R.K. Raina and J. Sokol, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris 353 (2015) 973–978.
[15] M.I.S. Robertson, On the theory of univalent functions, Ann. Math. 37 (1936) 374–408.
[16] M.S. Robertson, Certain classes of starlike functions, Michigan Math. J. 76 (1954) 755–758.
[17] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943) 48–82.
[18] K. Sharma, N.K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (2016) 923–939.
[19] J. Sokol, A certain class of starlike functions, Comput. Math. Appl. 62 (2011) 611–619.
[20] J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996) 101–105.
[21] S. Warschawski, On the higher derivatives at the boundary in conformal mappings, Trans. Amer. Math. Soc. 38 (1935) 310–340.
Volume 12, Issue 1
May 2021
Pages 847-855
  • Receive Date: 02 January 2018
  • Revise Date: 29 June 2018
  • Accept Date: 29 June 2018