Coefficient inequalities for analytic functions associated with cardioid domains

Abstract

The geometry of image domain of analytic functions is of substantial importance to have a comprehensive study of analytic functions. Malik et al. [Analytic functions associated with cardioid domain, submitted] introduced a new class of functions connected with cardioid domain and established coefficient bounds for functions in this class. Also the bounds for the coefficients of Taylor series and their related functional inequalities are of major interest. In this article, we aim to find the sharp bounds for the coefficients and to estimate the Fekete-Szegö functional for certain analytic functions associated with cardioid domain. The same type results are obtained for inverse functions and for $\log(f(z)/z)$.

Keywords

• analytic functions
• shell-like curve
• Fibonacci numbers
• cardioid domain
• starlike functions

References

1. [1] J. Dziok, R.K. Raina, and J. Sokół, Certain results for a class of convex functions related to shell-like curve connected with Fibonacci numbers, Comput. Math. Appl. 61, 2605–2613, 2011.
2. [2] J. Dziok, R.K. Raina, and J. Sokół, On $\alpha$-convex functions related to shell-like functions connected with Fibonacci numbers, Appl. Math. Comput. 218, 996–1002, 2011.
3. [3] J. Dziok, R.K. Raina, and J. Sokół, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Math. Comput. Modelling, 57, 1203–1211, 2013.
4. [4] M. Fekete and G. Szegö, Eine bemerkung uber ungerade schlichte funktionen, J. London Math. Soc. 8, 85–89, 1933.
5. [5] A.W. Goodman, Univalent Functions, Vol. I–II, Mariner Publishing Company, Tampa, Florida, USA, 1983.
6. [6] W. Janowski, Some extremal problems for certain families of analytic functions, Ann. Polon. Math. 28, 297–326, 1973.
7. [7] S. Kanas and A. Wiśniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105, 327–336, 1999.
8. [8] S. Kanas and A. Wiśniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45, 647–657, 2000.

9. [9] S.N. Malik, M. Raza, J. Sokół, and S. Zainab, Analytic functions associated with cardioid domain, submitted for publication.
10. [10] S.N. Malik, M. Raza, J. Sokół, and S. Zainab, On starlike functions associated with cardioid domain, submitted for publication.
11. [11] K.I. Noor and S.N. Malik, On a new class of analytic functions associated with conic domain, Comput. Math. Appl. 62, 367–375, 2011.
12. [12] K.I. Noor and S.N. Malik, On coefficient inequalities of functions associated with conic domains, Comput. Math. Appl. 62, 2209–2217, 2011.
13. [13] E. Paprocki and J. Sokół, The extremal problems in some subclass of strongly starlike functions, Folia Scient. Univ. Techn. Resoviensis, 157, 89–94, 1996.
14. [14] C. Pommerenke, Univalent Functions, Studia Math. Math. Lehrbucher, Vandenhoeck and Ruprecht, Göttingen, 1975.
15. [15] J. Sokół, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, 175, 111–116, 1999.