Bounds on the Third Order Hankel Determinant for Certain Subclasses of Analytic Functions

S.P. Vijayalakshmi; T.V. Sudharsan; Daniel Breaz; K.G. Subramanian

doi:10.1515/auom-2017-0045

.blurhash-client-img { display: none !important; }

Bounds on the Third Order Hankel Determinant for Certain Subclasses of Analytic Functions

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 25 (2017): Issue 3 (December 2017)

By: S.P. Vijayalakshmi, T.V. Sudharsan, Daniel Breaz and K.G. Subramanian

Open Access

|Mar 2018

Abstract

Let A be the class of analytic functions f(z) in the unit disc ∆ = {z ∈ C : |z| < 1g with the Taylor series expansion about the origin given by f(z) = z+ ∑_n=2^∞ a_nzⁿ, z ∈∆ : The focus of this paper is on deriving upper bounds for the third order Hankel determinant H₃(1) for two new subclasses of A.

References

[1] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications, S.S. Dragomir and J.Y. Cho, (Eds.), Nova Science Publishers, 6 (2010) 1-7.
Search in Google Scholar Back to article
[2] R. Bucur, D. Breaz and L. Georgescu, Third Hankel determinant for a class of analytic functions with respect to symmetric points, Acta Uni- versitatis Apulensis, 42 (2015) 79-86.10.17114/j.aua.2015.42.06
Search in Google Scholar Back to article
[3] D. Banasal, S. Maharana and J. K. Prajapat, Third Order Hankel deter- minant for certain univalent functions, J. Korean Math.Soc. 52(6) (2015) 1139-1148.10.4134/JKMS.2015.52.6.1139
Search in Google Scholar Back to article
[4] P.L. Duren, Univalent functions, Springer Verlag, New York Inc., 1983.
Search in Google Scholar Back to article
[5] R.M. Goel and B.S. Mehrok, A class of univalent functions, J. Austral. Math. Soc. (Series A) 35 (1983) 1-17.10.1017/S1446788700024733
Search in Google Scholar Back to article
[6] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. Journal of Math. Analysis, 4(52) (2010) 2473-2585.
Search in Google Scholar Back to article
[7] A. Janteng, S. Abdulhalim and M. Darus, Coeficient inequality for a function whose derivative has positive real part, J. Inequalities Pure and Applied Math. 7(2) Art. 50 (2006) 1-5.
Search in Google Scholar Back to article
[8] A. Janteng, S. Abdulhalim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1(13) (2007) 619-625.
Search in Google Scholar Back to article
[9] A. Janteng, S.A. Halim and M. Darus, Estimate on the second Hankel functional for functions whose derivative has a positive real part, Journal of Quality Measurement and Analysis, 4(1) (2008) 189-195.
Search in Google Scholar Back to article
[10] D. V. Krishna, B. Venkateswarlu, T. RamReddy, Third Hankel determi- nant for bounded turning functions of order alpha, J. Nigerian Mathe- matical Society, 34 (2015) 121-127. 10.1016/j.jnnms.2015.03.001
Search in Google Scholar Back to article
[11] D. V. Krishna, T. RamReddy, Coeficient inequality for a function whose derivative has a positive real part of order ; Mathematica Bohemica 140(1) (2015) 43-52.10.21136/MB.2015.144178
Search in Google Scholar Back to article
[12] N. Kharudin, A. Akbarally, D. Mohamad and S.C. Soh, The second Han- kel determinant for the class of close to convex functions, European J. Scientific Research, 66(3) (2011) 421-427.
Search in Google Scholar Back to article
[13] R.J. Libera and E.J. Zlotkiewiez, Early coeficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85(2) (1982) 225-230.10.1090/S0002-9939-1982-0652447-5
Search in Google Scholar Back to article
[14] R.J. Libera and E.J. Zlotkiewiez, Coeficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2) (1983) 251-257.10.1090/S0002-9939-1983-0681830-8
Search in Google Scholar Back to article
[15] T.H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962) 532-537.10.1090/S0002-9947-1962-0140674-7
Open DOI Search in Google Scholar Back to article
[16] A.K. Mishra, J.K. Prajapat and S. Maharana, Bounds on Hankel determi- nant for starlike and convex functions with respect to symmetric points, Cogent Mathematics, 3: 1160557 (2016). [17] J.W. Noonan and D.K. Thomas. On the second Hankel determinant of areally mean p-valent functions. Trans. Amer. Math. Soc.,223(2)(1976) 337-34610.1080/23311835.2016.1160557
Search in Google Scholar Back to article
[18] K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures et Appl., 28(c) (1983) 731-739.
Search in Google Scholar Back to article
[19] M.S. Robertson, On the theory of univalent functions, Annals of Math., 37 (1936) 374-408.10.2307/1968451
Search in Google Scholar Back to article
[20] C. Selvaraj and N. Vasanthi, Coeficient bounds for certain subclasses of close-to-convex functions, Int. J. Math. Analysis, 4(37) (2010) 1807-1814.
Search in Google Scholar Back to article
[21] G. Shanmugam, B. A. Stephen and K.G. Subramanian, Second Hankel determinant for certain classes of analytic functions, Bonfring Int. J. Data Mining, 2(2) (2012) 57-60.10.9756/BIJDM.1368
Search in Google Scholar Back to article
[22] T.V. Sudharsan, S.P. Vijayalakshmi, B.A. Stephen, Third Hankel deter- minant for a subclass of analytic univalent functions, Malaya J. Matem- atik, 2(4)(2014) 438-444.10.26637/mjm204/011
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/auom-2017-0045 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

Journal RSS Feed

Language: English

Page range: 199 - 214

Submitted on: Sep 9, 2016

Accepted on: Oct 24, 2016

Published on: Mar 31, 2018

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

Analytic functions,

Hankel determinant,

Functions with positive real part

Related subjects:

Mathematics,

General mathematics

© 2018 S.P. Vijayalakshmi, T.V. Sudharsan, Daniel Breaz, K.G. Subramanian, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 25 (2017): Issue 3 (December 2017)