On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain
R., S., R., E., K.G., S., T.V., S. (2025). On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain. EKB Journal Management System, 16(2), 1-11. doi: 10.21608/jfca.2025.353987.1154
SATHISH SRINIVASAN R.; Ezhilarasi R.; Subramanian K.G.; Sudharsan T.V.. "On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain". EKB Journal Management System, 16, 2, 2025, 1-11. doi: 10.21608/jfca.2025.353987.1154
R., S., R., E., K.G., S., T.V., S. (2025). 'On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain', EKB Journal Management System, 16(2), pp. 1-11. doi: 10.21608/jfca.2025.353987.1154
R., S., R., E., K.G., S., T.V., S. On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain. EKB Journal Management System, 2025; 16(2): 1-11. doi: 10.21608/jfca.2025.353987.1154

On the second and third order Vandermonde determinants for a subclass of analytic functions in the Limaçon domain

Journal of Fractional Calculus and Applications
Volume 16, Issue 2, 2025, Page 1-11  XML PDF (745.87 K)
Document Type: Regular research papers
DOI: 10.21608/jfca.2025.353987.1154
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Authors
SATHISH SRINIVASAN R. email 1; Ezhilarasi R.2; Subramanian K.G.3; Sudharsan T.V.4
1Research Scholar, Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai-600073, Tamilnadu, India.
2Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai-600073, Tamilnadu, India.
3Honorary Visiting Professor, Department of Mathematics and Computer Science, Faculty of Science, Liverpool Hope University, Liverpool L16 9JD, UK.
4(Retired), Department of Mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai-600073, Tamilnadu, India.
Abstract
A Limaçon curve is defined by ∂L(p)={a+ib∈C∶[(a-1)^2+b^2-p^4 ]^2=4p^2 [(a-1+p^2 )^2+b^2]} where p∈[-1,1]∖{0}. A Limaçon curve also known as Limaçon of Pascal has many applications in the field of mathematics, physics, engineering and fluid dynamics. Vandermonde determinants are used in linear algebra, optimization and frequency analysis. Motivated by this, in this paper we define a new subclass of analytic functions related to Limaçon domain. Let TL_p (α),0≤α≤1,0<p≤1/√2, denote the subclass of normalized analytic functions f(z)=z+∑_(r=2)^∞▒〖a_r z^r 〗 in the open unit disk U={z∈C∶|z|<1} satisfying the condition
(2(αz^2 f^'' (z)+zf^' (z)))/(αz(f(z)-f(-z))^'+(1-α)(f(z)-f(-z)))≺L_p (z) z∈U,
where L_p (z)=(1+pz)^2 is the Limaçon function and ≺ denotes the well-known subordination of functions in geometric function theory. In this paper, we determine the sharp coefficient bounds for the second order Vandermonde determinants and upper bounds for the third order Vandermonde determinants for functions in the subclass TL_p (α). Further, we obtain as corollaries the results of already known classes.
Keywords
Vandermonde determinant; analytic function; Limaçon; subordination
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