On certain geometric properties of generalized polylogarithm function

V. Agnes Sagaya Judy Lavanya; M.P. Jeyaraman; H. Aaisha Farzana

doi:10.26637/MJM0801/0031

Abstract

In this manuscript, we investigate the Hadamard product $H_f(a, b ; z)$ of normalized analytic functions in the unit disc $\Delta$ and generalized second order polylogarithm function $G(a, b ; z)$, where
$$
G(a, b ; z)=\sum_{n=1}^{\infty} \frac{(a+1)(b+1)}{(n+a)(n+b)} z^n, a, b \in \mathbb{C} \backslash\{-1,-2, \ldots\} .
$$
Further, we derive certain characteristics of the function $H_f(a, b ; z)$ and obtain various sufficient conditions for the function $H_f(a, b ; z)$ to be Janowski starlike. Also certain inequalities containing the function $H_f(a, b ; z)$ are obtained.

Keywords:

Analytic functions, Convolution, Subordination, Generalized polylogarithm function

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

V. Agnes Sagaya Judy Lavanya Department of Mathematics, Dr. MGR Janaki College of Arts and Science, Chennai -600028, Tamil Nadu, India.
M.P. Jeyaraman Department of Mathematics, L.N. Government College, Ponneri-601204, Tamil Nadu, India.
H. Aaisha Farzana Department of Mathematics, A.M. Jain College, Meenambakkam, Chennai-600114, Tamil Nadu, India. https://orcid.org/0000-0003-4195-3708

Pages: 183-188
Date Published: 01-01-2020
Vol. 8 No. 01 (2020): Malaya Journal of Matematik (MJM)

B.C. Berndt, Ramanujan's Notebook, Part I, SpringerVerlag, Berlin-Heidelberg, New York, 1989.

H. Silverman, Subclasses of starlike functions, Revue Roumaine de MathematiquePures et Appliquees, 23(1978), 1093-1099.

I.S. Jack, Functions starlike and convex of order, J. Lond. Math. Soc., 2(3)(1971), 469-474.

L. Lewin, Polylogarithms and Associated Functions, Elsevier North-Holland, New York and Oxford, 1981.

T.H. MacGregor, The radius of univalence of certain analytic functions, II. Proc. Amer. Math. Soc., 14(1963), 521-524.

M. Nunokawa, On some angular estimate of analytic functions, Math. Japon, 41(2)(1995), 447-452.

M. Obradovic and N. Tuneski, On the starlike criteria defined by Silverman, ZeszytyNaukowePolitechnikiRzeszowskiejMatematyka, 24(2000), 59-64.

S. Ponnusamy, Differential subordination and starlike functions, Complex Variables Theory Appl., 19(1992), $185-194$.

S. Ponnusamy, Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a half plane, Rocky Mountain Journal of Mathematics, 28(1998), 695-732.

V. Ravichandran and K. Sharma, Sufficient conditions for starlikeness, J. Korean Math. Soc., 52(2015), 727-749.

S.S. Miller and P.T. Mocanu, Differential Subordinations, Monographs and Textbook in Pure and Applied Mathematics, 225, Dekker, New York, 2000.

S.S. Miller and P.T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl., $65(1978), 289-305$.

K. Sharma and V. Ravichandran, Sufficient conditions for Janowskistarlike functions, Stud. Univ. Babes-Bolyai Math., 61(2016), 63-76.

N. Tuneski and M. Darus, On functions that are Janowskistarlike with respect to N-symmetric points, Hacettepe Journal of Mathematics and Statistics, 41(2012), 271275.