Third Hankel determinant for certain subclass of analytic functions

M. A. Ganiyu; K.O. Babalola

doi:10.26637/mjm404/005

Abstract

The third Hankel determinant, $H_3(1)$ for subclass of analytic functions satisfying geometric condition
$$
\operatorname{Re} \frac{z f^{\prime}(z)}{f(z)} \frac{f(z)^{\alpha-1} f^{\prime}(z)}{z^{\alpha-1}}>0
$$
for nonnegative real number $\alpha$, in the open unit disk $U=\{z \in \mathbb{C}:|z|<1\}$ is derived in line with a method of classical analysis devised by Libera and Zlotkiewicz [9].

Keywords:

Hankel determinant, caratheodory functions, product of geometric expression, analytic functions

Mathematics Subject Classification:

30C45

Authors Imprint References Funding Related Articles Downloads

M. A. Ganiyu Department of Physical Sciences, College of Natural Sciences, Al-Hikmah University, Ilorin, Nigeria.
K.O. Babalola Department of Mathematics, College of science, University of Ilorin, Ilorin, Nigeria.

Pages: 565-570
Date Published: 01-10-2016
Vol. 4 No. 04 (2016): Malaya Journal of Matematik (MJM)

Abubaker, A. and Darus, M., Hankel determinant for a class of analytic functions involving a generalized linear differential operator, International Journal of Pure and Applied Mathematics, 69 (4) (2011), 429-435.

Al-Refai, O. and Darus, M., Second Hankel determinant for a class of analytic functions defined by a fractional operator, European Journal of Science Resources , 28 (2) (2009), 234-241.

Babalola, K.O. and Opoola, T.O. , On the coefficients of certain analytic and univalent functions, Advances in Inequalities for Series, Nova Science Publishers, (2008), 5-17.

Babalola, K.O., On $mathrm{H}_3(1)$ Hankel Determinant for some Classes of Univalent Functions, Inequality Theory and Applications, Nova Science Publishers, Inc., 6 (2010), 1-7.

Duren, P.L., Univalent functions , Mariner Publishing Co. Inc. Tampa, Florida , 1 (1983) .

Ganiyu, M.A., Jimoh, F.M., Ejieji, C.N. and Babalola, K.O., Second Hankel determinant for Certain Subclasses of Analytic functions, Submitted.

Janteng, A., Abdulhalim, S. and Darus, M., Hankel determinant for starlike and convex functions, Int. Journal of Math. Analysis, 1 (13) (2007), 619-625.

Jimoh, F.M., Ganiyu, M.A., Ejieji, C.N., and Babalola, K.O., Bounds on coefficients of certain Analytic functions, Nigeria Journal of Pure and Applied Science, Faculty of Science, University of Ilorin, 26 (2013), 24112418.

Libera, R.J and Zlotkiewicz, E.J., Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (2) (1983), 251-257. DOI: https://doi.org/10.1090/S0002-9939-1983-0681830-8

Noonan, J.W. and Thomas, D.K., Second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337-346. DOI: https://doi.org/10.1090/S0002-9947-1976-0422607-9

Norlyda, M., Daud, M. and Shaharuddin, C.S., Second Hankel determinant for certain generalized classes of analytic functions, International Journal of Mathematical Analysis, 6 (17) (2012), 807-812.

Shanmugam, G., Adolf, Stephen B. and Babalola, K.O., Third Hankel determinant for $alpha$-starlike functions, Gulf Journal of Mathematics, 2 (2) (2014), 107-113. DOI: https://doi.org/10.56947/gjom.v2i2.202

Vamshee, D.K. and Ramreddy, T. , Hankel determinant for p-valent starlike and convex functions of order $alpha$, Novi sad J. Math., 42 (2) (2012),89-96.

Vamshee, D. K., Venkateswarlu, B. and Ramreddy T., Third Hankel determinant for bounded turning functions of order alpha, Journal of the Nigerian Mathematical Society, 34 (2) (2015) 121-127. DOI: https://doi.org/10.1016/j.jnnms.2015.03.001