Coefficient inequality for transforms of parabolic starlike and uniformly convex functions
Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 39-56.

The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the k th root transform f(z k ) 1 k of normalized analytic function f(z) belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

DOI : 10.5802/ambp.341
Classification : 30C45, 30C50
Mots clés : Analytic function, parabolic starlike and uniformly convex functions, upper bound, second Hankel functional, positive real function, Toeplitz determinants.
Vamshee Krishna, D. 1 ; Venkateswarlu, B. 1 ; RamReddy, T. 2

1 Department of Mathematics GIT, GITAM University Visakhapatnam- 530 045, A.P., India.
2 Department of Mathematics, Kakatiya University, Warangal- 506 009, A.P., India.
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Vamshee Krishna, D.; Venkateswarlu, B.; RamReddy, T. Coefficient inequality for transforms of parabolic starlike and uniformly convex functions. Annales mathématiques Blaise Pascal, Tome 21 (2014) no. 2, pp. 39-56. doi : 10.5802/ambp.341. http://www.numdam.org/articles/10.5802/ambp.341/

[1] Abubaker, A.; Darus, M. Hankel Determinant for a class of analytic functions involving a generalized linear differential operators, Int. J. Pure Appl. Math., Volume 69(4) (2011), pp. 429 -435 (MR 2847841 | Zbl 1220.30011) | MR | Zbl

[2] Alexnader, J. W. Functions which map the interior of the unit circle upon simple regions, Annal. of. Math., Volume (2)17 (1915), pp. 12 -22 (MR 1503516 | JFM 45.0672.02) | DOI | MR

[3] Ali, R. M. Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc.(second series), Volume 26(1) (2003), pp. 63 -71 MR 2055766 (2005b:30011) | Zbl 1185.30010. | MR | Zbl

[4] Ali, R. M. Starlikeness associated with parabolic regions, Int. J. Math. Math. Sci., Volume 4 (2005), pp. 561-570 MR 2172395 (2006d:30011) | Zbl 1077.30011. | DOI | MR | Zbl

[5] Ali, R. M.; Lee, S. K.; Ravichandran, V.; Supramaniam, S. The Fekete-Szego ¨ coefficient functional for transforms of analytic functions, Bull. Iran. Math. Soc., Volume 35(2) (2009), pp. 119-142 (MR 2642930.) | MR | Zbl

[6] Ali, R. M.; Singh, V. Coefficients of parabolic starlike functions of order ρ, World Sci. Publ. River Edge, New Jersey, 1995, pp. 23 -36 (1995 MR 1415158 [97 h:30008].) | MR | Zbl

[7] Duren, P. L. Univalent functions, 259, Grundlehren der Mathematischen Wissenschaften, New York, Springer-verlag XIV, 328, 1983 (MR 0708494 | Zbl 0514.30001) | MR | Zbl

[8] Ehrenborg, R. The Hankel determinant of exponential polynomials, Amer. Math. Monthly, Volume 107(6) (2000), pp. 557-560 MR 1767065 (2001c:15009) | Zbl 0985.15006. | DOI | MR | Zbl

[9] Goodman, A. W. On uniformly convex functions, Ann. Polon. Math., Volume 56 (1) (1991), pp. 87 -92 MR 1145573 (93a:30009) | Zbl 0744.30010. | MR | Zbl

[10] Grenander, U.; Szegő, G. Toeplitz forms and their applications, Second edition. Chelsea Publishing Co., New York, 1984 (MR 0890515 | Zbl 0611.47018) | MR | Zbl

[11] Janteng, A.; Halim, S. A.; Darus, M. Coefficient inequality for a function whose derivative has a positive real part, J. Inequl. Pure Appl. Math., Volume 7(2) (2006), pp. 1-5 (MR 2221331 | Zbl 1134.30310.) | MR | Zbl

[12] Janteng, A.; Halim, S. A.; Darus, M. Hankel determinant for starlike and convex functions, Int. J. Math. Anal., (Ruse), Volume 4 (no. 13-16) (2007), pp. 619-625 (MR 2370200 | Zbl 1137.30308.) | MR | Zbl

[13] Layman, J. W. The Hankel transform and some of its properties, J. Integer Seq., Volume 4 (1) (2001), pp. 1-11 (MR 1848942 | Zbl 0978.15022.) | MR | Zbl

[14] Libera, R. J.; Zlotkiewicz, E. J. Coefficient bounds for the inverse of a function with derivative in , Proc. Amer. Math. Soc., Volume 87 (1983), pp. 251-257 (MR 0681830 | Zbl 0488.30010.) | MR | Zbl

[15] Ma, W. C.; Minda, D. Uniformly convex functions, Ann. Polon. Math., Volume 57(2) (1992), pp. 165 -175 (MR 1182182.) | MR | Zbl

[16] Noonan, J. W.; Thomas, D. K. On the second Hankel determinant of areally mean p - Valent functions, Trans. Amer. Math. Soc., Volume 223(2) (1992), pp. 337 -346 (MR 0422607 | Zbl 0346.30012) | MR | Zbl

[17] Noor, K. I. Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., Volume 28(8) (1983), pp. 731 -739 (MR 0725316 | Zbl 0524.30008.) | MR | Zbl

[18] Pommerenke, Ch. On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc., Volume 41 (1966), pp. 111-122 (MR 0185105 | Zbl 0138.29801.) | DOI | MR | Zbl

[19] Pommerenke, Ch. Univalent functions, Vandenhoeck and Ruprecht, Gottingen, 1975 (MR 0507768 | Zbl 0298.30014.) | MR | Zbl

[20] Ronning, F. A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie - Sklodowska Sect. A., Volume 47 (1993), pp. 123 -134 (MR 1344982 | Zbl 0879.30004.) | MR | Zbl

[21] Simon, B. Orthogonal polynomials on the unit circle, Part 1. Classical theory, AMS Colloquium Publ. 54, Part 1, American Mathematical Society, Providence, RI, 2005 (MR 2105088| Zbl 1082.42020) | MR | Zbl

[22] VamsheeKrishna, D.; RamReddy, T. Coefficient inequality for uniformly convex functions of order α, J. Adv. Res. Pure Math., Volume 5(1) (2013), pp. 25-41 (MR 3020966.) | DOI | MR

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