Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions

Volume 18, Issue 4, pp 453--459 Publication Date: December 12, 2018       Article History
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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.

Keywords

• Hankel determinant
• modified Mittag-Leffler function
• polylogarithms functions

•  30C45
•  30C50

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.