Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions
Volume 18, Issue 4, pp 453--459
http://dx.doi.org/10.22436/jmcs.018.04.06
Publication Date: December 12, 2018
Submission Date: January 26, 2018
Revision Date: October 30, 2018
Accteptance Date: November 16, 2018
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Authors
M. N. M. Pauzi
- School of Modelling and Data Science (Previously: School of Mathematical Sciences), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D.E., Malaysia.
M. Darus
- School of Modelling and Data Science (Previously: School of Mathematical Sciences), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D.E., Malaysia.
S. Siregar
- Department of Science and Biotechnology, Faculty of Engineering and Life Sciences, Universiti Selangor, Batang Berjuntai, Bestari, Jaya 45600, Selangor D.E., Malaysia.
Abstract
In this work, a new generalized derivative operator \( \mathfrak{M}_{\alpha,\beta,\lambda}^{m}\) is introduced. This operator obtained by using convolution (or Hadamard product) between the linear operator of the generalized Mittag-Leffler function in terms of the extensively-investigated Fox-Wright \(_{p}\Psi_{q}\) function and generalized polylogarithm functions defined by
\[
\mathfrak{M}_{\alpha,\beta,\lambda}^{m}f(z)=\mathfrak{F}_{\alpha,\beta}f(z)*\mathfrak{D}_{\lambda}^{m}f(z)
= z+\sum_{n=2}^{\infty}\frac{\Gamma(\beta)n^{m}(n+\lambda-1)!}{\Gamma[\alpha(n-1)+\beta]\lambda ! (n-1)!}a_{n}z^{n},
\]
where \(m \in \mathbb{N}_{0} = \{0,1,2,3,\ldots\}\) and \(\min\{Re(\alpha),Re(\beta)\}>0\). By making use of \(\mathfrak{M}_{\alpha,\beta,\lambda}^{m}f(z)\), a class of analytic functions is introduced. The sharp upper bound for the nonlinear \(|a_{2}a_{4}-a_{3}^{2}|\) (also called the second Hankel functional) is obtained. Relevant connections of the results presented here with those given in earlier works are also indicated.
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ISRP Style
M. N. M. Pauzi, M. Darus, S. Siregar, Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions, Journal of Mathematics and Computer Science, 18 (2018), no. 4, 453--459
AMA Style
Pauzi M. N. M., Darus M., Siregar S., Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions. J Math Comput SCI-JM. (2018); 18(4):453--459
Chicago/Turabian Style
Pauzi, M. N. M., Darus, M., Siregar, S.. "Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions." Journal of Mathematics and Computer Science, 18, no. 4 (2018): 453--459
Keywords
- Hankel determinant
- modified Mittag-Leffler function
- polylogarithms functions
MSC
References
-
[1]
A. Abubaker, M. Darus, Hankel determinant for a class of analytic functions involving a generalized linear differential operator, Int. J. Pure Appl. Math., 69 (2011), 429–435.
-
[2]
M. H. Al-Abbadi, M. Darus , Hankel Determinant for certain class of analytic function defined by generalized derivative operator , Tamkang J. Math., 43 (2012), 445–453.
-
[3]
O. Al-Refai, M. Darus, Second Hankel determinant for a class of analytic functions Defined by a fractional operator , European Journal of Scientific Research, 28 (2009), 234–241.
-
[4]
K. Al Shaqsi, M. Darus, An operator defined by convolution involving the polylogarithms functions, J. Math. Stat., 4 (2008), 46–50.
-
[5]
D. Bansal , Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., 26 (2013), 103–107.
-
[6]
D. G. Cantor, Power series with integral coefficients, Bull. Amer. Math. Soc., 26 (1963), 362–366.
-
[7]
R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557–560.
-
[8]
M. Fekete, G. Szegö, Eine Bemerkung uber ungerade schlichte funktionen, J. London Math. Soc., 8 (1933), 85–89.
-
[9]
U. Grenander, G. Szegö, Toeplitz Forms and their Application, University of California Press, Berkeley–Los Angeles (1958)
-
[10]
A. Janteng, S. A. Halim, M. Darus, Coefficient Inequality for a Function whose Derivative has a Positive Real Part, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006), 5 pages.
-
[11]
A. Janteng, S. A. Halim, M. Darus, Hankel Determinant for Starlike and Convex Functions, Int. J. Math. Anal. (Ruse), 1 (2007), 619–625.
-
[12]
F. R. Keogh, E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8–12.
-
[13]
J. W. Layman , The Hankel transform and some of its properties , J. Integer Seq., 4 (2001), 11 pages.
-
[14]
R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251–289.
-
[15]
R. J. Libera, E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85 (1982), 225–230.
-
[16]
J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean and p-valent function, Trans. Amer. Math. Soc., 223 (1976), 337–346.
-
[17]
K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine Math. Pures Appl., 28 (1983), 731–739.
-
[18]
K. I. Noor, S. A. Al-Bany, On Bazilevic functions, Internat. J. Math. Math. Sci., 10 (1987), 79–88.
-
[19]
C. H. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen (1975)
-
[20]
H. M. Srivastava, A. Kilicman, Z. E. Abdulnaby, R. W. Ibrahim, Generalized convolution properties based on the modified Mittag-Leffler function, J. Nonlinear Sci. Appl., 10 (2017), 4284–4294.
-
[21]
R. Wilson, Determinantal criteria for meromorphic functions, Proc. London Math. Soc., 4 (1954), 357–374.
