Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Turkish Journal of Analysis and Number Theory. 2015, 3(6), 149-153
DOI: 10.12691/tjant-3-6-2
Open AccessArticle

Upper Bound of Partial Sums Determined by Matrix Theory

Rabha W. Ibrahim1,

1Institute of Mathematical Sciences, University Malaya, Malaysia,

Pub. Date: December 23, 2015

Cite this paper:
Rabha W. Ibrahim. Upper Bound of Partial Sums Determined by Matrix Theory. Turkish Journal of Analysis and Number Theory. 2015; 3(6):149-153. doi: 10.12691/tjant-3-6-2


One of the major problems in the geometric function theory is the coefficients bound for functional and partial sums. The important method, for this purpose, is the Hankel matrix. Our aim is to introduce a new method to determine the coefficients bound, based on the matrix theory. We utilize various kinds of matrices, such as Hilbert, Hurwitz and Turan. We illustrate new classes of analytic function in the unit disk, depending on the coefficients of a particular type of partial sums. This method shows the effectiveness of the new classes. Our results are applied to the well known classes such as starlike and convex. One can illustrate the same method on other classes.

analytic function univalent function unit disk partial sums coefficients bound

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/


[1]  P. Dienes, The Taylor Series. Dover, New York (1957).
[2]  A. Edrei, Sur les dterminants rcurrents et les singularits dune fonction done por son dveloppement de Taylor. Compos. Math. 7, 20-88 (1940).
[3]  D. G. Cantor, Power series with integral coefficients. Bull. Am. Math. Soc. 69, 362-366 (1963).
[4]  R. Wilson, Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc. 4, 357-374 (1954).
[5]  R. Vein, P. Dale, Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, vol. 134. Springer, New York (1999).
[6]  D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett. 26(1), 103-107 (2013).
[7]  K. O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 6, 1-7 (2007).
[8]  R. W. Ibrahim, Bounded nonlinear functional derived by the generalized Srivastava-Owa fractional differential operator. International Journal of Analysis, 1-7 (2013).
[9]  G. Szego, Zur theorie der schlichten abbilungen. Math. Ann. 100, 188-211 (1928).
[10]  S. Owa, Partial sums of certain analytic functions. Int. J. Math. Math. Sci. 25(12), 771-775 (2001).
[11]  M. Darus, R. W. Ibrahim, Partial sums of analytic functions of bounded turning with applications. Comput. Appl. Math. 29(1), 81-88 (2010).
[12]  R. W. Ibrahim, M. Darus, Ces├íro partial sums of certain analytic functions, Journal of Inequalities and Applications, 51, 1-9 (2013).