Kumar, S., Ravichandran, V., Verma, S. (2017). Initial coefficients of starlike functions with real coefficients. Bulletin of the Iranian Mathematical Society, 43(6), 1837-1854.

S. Kumar; V. Ravichandran; S. Verma. "Initial coefficients of starlike functions with real coefficients". Bulletin of the Iranian Mathematical Society, 43, 6, 2017, 1837-1854.

Kumar, S., Ravichandran, V., Verma, S. (2017). 'Initial coefficients of starlike functions with real coefficients', Bulletin of the Iranian Mathematical Society, 43(6), pp. 1837-1854.

Kumar, S., Ravichandran, V., Verma, S. Initial coefficients of starlike functions with real coefficients. Bulletin of the Iranian Mathematical Society, 2017; 43(6): 1837-1854.

Initial coefficients of starlike functions with real coefficients

^{1}Bharati Vidyapeeth's College of Engineering, Delhi--110063, India.

^{2}Department of Mathematics, University of Delhi Delhi--110 007, India.

^{3}Department of Mathematics, University of Delhi, Delhi--110 007, India.

Receive Date: 12 May 2016,
Revise Date: 26 October 2016,
Accept Date: 26 October 2016

Abstract

The sharp bounds for the third and fourth coefficients of Ma-Minda starlike functions having fixed second coefficient are determined. These results are proved by using certain constraint coefficient problem for functions with positive real part whose coefficients are real and the first coefficient is kept fixed. Analogous results are obtained for a general class of close-to-convex functions

H.S. Al-Amiri and D. Bshouty, Constraint coefficient problems for subclasses of univalent functions, in: Current Topics in Analytic Function Theory, pp. 29--35, World Scientific Publ. River Edge, NJ, 1992.

H.S. Al-Amiri and D.H. Bshouty, A constraint coefficient problem with an application to a convolution problem, Complex Variables Theory Appl. 22 (1993), no. 3-4, 241--246.

R.M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc. (2) 26 (2003), no. 1, 63--71.

R.M. Ali, S.K. Lee, V. Ravichandran and S. Supramaniam, The Fekete-Szeg}o coefficient functional for transforms of analytic functions, Bull. Iranian Math. Soc. 35 (2009), no. 2, 119--142, 276.

R.M. Ali, V. Ravichandran and N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. 187 (2007), no. 1, 35--46.

M.F. Ali and A. Vasudevarao, Coefficient inequalities and Yamashita's conjecture for some classes of analytic functions, J. Aust. Math. Soc. 100 (2016), no. 1, 1--20.

K.O. Babalola, The fifth and sixth coefficients of α-close-to-convex functions, Kragujevac J. Math. 32 (2009) 5--12.

L.E. Dubins, On extreme points of convex sets, J. Math. Anal. Appl. 5 (1962) 237--244.

P.L. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer-Verlag, New York, 1983.

D.J. Hallenbeck and T.H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman, Boston, MA, 1984.

S.G. Hamidi and J.M. Jahangiri, Faber polynomial coefficient estimates for bi-univalent functions defined by subordinations, Bull. Iranian Math. Soc. 41 (2015), no. 5, 1103--1119.

Z.J. Jakubowski, H. Siejka and O. Tammi, On the maximum of a4 3a2a3 +μa2 and some related functionals for bounded real univalent functions, Ann. Polon. Math. 46 (1985) 115--128.

W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 28 (1973) 297--326.

S. Kanas and A. Tatarczak, Constrained coefficients problem for generalized typically real functions, Complex Var. Elliptic Equ. 61 (2016), no. 8, 1052--1063.

P. Koulorizos and N. Samaris, The Landau problem for nonvanishing functions with real coefficients, J. Comput. Appl. Math. 139 (2002), no. 1, 129--139.

S. Kumar and V. Ravichandran, A subclass of starlike functions associated with a rational function, Southeast Asian Bull. Math. 40 (2016) 199-212.

A. Lecko, On coefficient inequalities in the Caratheodory class of functions, Ann. Polon. Math. 75 (2000), no. 1, 59--67.

Z. Lewandowski, J. Miazga and J. Szynal, Koebe domains for univalent functions with real coefficients under Montel's normalization, Ann. Polon. Math. 30 (1975), no. 3, 333--336.

R.J. Libera and E.J.Z lotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), no. 2, 225--230.

A.E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Amer Math. Soc. 21 (1969) 545--552.

W.C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), pp.

157--169, Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994.

M.T. McGregor, On three classes of univalent functions with real coefficients, J. Lond. Math. Soc. 39 (1964) 43--50.

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), no. 1, 365--386.

S.S. Miller and P.T. Mocanu, Differential Subordinations, Marcel Dekker, New York, 2000.

M. Nunokawa, S. Owa, J. Nishiwaki and H. Saitoh, Sufficient conditions for starlikeness and convexity of analytic functions with real coefficients, Southeast Asian Bull. Math. 33 (2009), no. 6, 1149--1155.

V. Ravichandran, Starlike and convex functions with respect to conjugate points, Acta Math. Acad. Paedagog. Nyhazi. (N.S.) 20 (2004), no. 1, 31--37.

V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353 (2015), no. 6, 505--510.

N. Samaris, Constrained coefficient problems of certain classes of analytic functions, Analysis (Munich) 24 (2004), no. 3, 197--211.

N. Samaris and P. Koulorizos, Constraint coefficient problems for a subclass of starlike functions, Publ. Math. Debrecen 56 (2000), no. 1-2, 63--76.

T.N. Shanmugam, C. Ramachandran and V. Ravichandran, Fekete-Szeg}o problem for subclasses of starlike functions with respect to symmetric points, Bull. Korean Math. Soc. 43 (2006), no. 3, 589--598.

T.N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, in: Computational Methods and Function Theory 1994 (Penang), pp. 319--324,

Ser. Approx. Decompos. 5, World Scientific Publ. River Edge, NJ, 1995.

M. Sobczak-Knec, Koebe domains for certain subclasses of starlike functions, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 61 (2007) 129--135.

M. Sobczak-Knec and P. Zaprawa, Covering domains for classes of functions with real coefficients, Complex Var. Elliptic Equ. 52 (2007), no. 6, 519--535.

J. Sokol, Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J. 49 (2009), no. 2, 349--353.

J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996) 101--105.