[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.5666/KMJ.2018.58.2.231

Sharp Coefficient Bounds for the Quotient of Analytic Functions

Park, Ji Hyang (Department of Applied Mathematics, Pukyong National University)
Kumar, Virendra (Department of Applied Mathematics, Pukyong National University)
Cho, Nak Eun (Department of Applied Mathematics, Pukyong National University)

Publication Information

Kyungpook Mathematical Journal / v.58, no.2, 2018 , pp. 231-242 More about this Journal

Abstract

We derive sharp upper bound on the initial coefficients and Hankel determinants for normalized analytic functions belonging to a class, introduced by Silverman, defined in terms of ratio of analytic representations of convex and starlike functions. A conjecture related to the coefficients for functions in this class is posed and verified for the first five coefficients.

Keywords

univalent function; coefficient bound; Hankel determinant;

Citations & Related Records

Reference

1	M. Obradowic and N. Tuneski, On the starlike criteria defined by Silverman, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 24(2000), 59-64.
2	R. Ohno and T. Sugawa, Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions, Kyoto J. Math., 58(2)(2018), 227-241. DOI
3	Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc., 41(1966), 111-122.
4	D. V. Prokhorov and J. Szynal, Inverse coefficients for ( ${\alpha}$ , ${\beta}$ )-convex functions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 35(1981), 125-143.
5	V. Ravichandran, Y. Polatoglu, M. Bolcal and A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacet. J. Math. Stat., 34(2005), 9-15.
6	V. Ravichandran and S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris, 353(6)(2015), 505-510. DOI
7	H. Silverman, Convex and starlike criteria, Int. J. Math. Math. Sci., 22(1)(1999), 75-79. DOI
8	N. Tuneski, On the quotient of the representations of convexity and starlikeness, Math. Nachr., 248/249(2003), 200-203. DOI
9	M. Nunokawa, S. Owa, H. Saitoh and O. P. Ahuja, On the quotient of the analytic representations of convex and starlike functions, Surikaisekikenkyusho Kokyuroku, 1112(1999), 63-69.
10	R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2)(1983), 251-257. DOI
11	R. M. Ali, N. E. Cho, V. Ravichandran and S. S. Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math., 16(3)(2012), 1017-1026. DOI
12	C. Caratheodory, Uber den variabilitatsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen, Math. Ann., 64(1)(1907), 95-115. DOI
13	C. Caratheodory, Uber den variabilitatsbereich der fourier'schen konstanten von positiven harmonischen funktionen, Rend. Circ. Mat. Palermo, 32(1911), 193-217. DOI
14	W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc., 18(3)(1968), 77-94.
15	A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. (Ruse), 1(13-16)(2007), 619-625.
16	F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20(1969), 8-12. DOI
17	R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(2)(1982), 225-230. DOI
18	S. S. Kumar, V. Kumar, V. Ravichandran and N. E. Cho, Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl., (2013), Art. ID 2013:176, 13 pp.
19	S. K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl., (2013), Art. ID. 2013:281, 17 pp.