Mathematics and Statistics 11(2): 345-352, 2023 DOI: 10.13189/ms.2023.110214 http://www.hrpub.org Toeplitz Determinant For Error Starlike & Error Convex Function D Kavitha 1 , K Dhanalakshmi 2,* , K Anitha 1 1 Department of Mathematics, SRM Institute of Science and Technology, India 2 Department of Mathematics, Theivanai Ammal College for Women, India Received November 6, 2022; Revised January 15, 2023; Accepted January 28, 2023 Cite This Paper in the following Citation Styles (a): [1] D Kavitha, K Dhanalakshmi, K Anitha, ”Toeplitz Determinant For Error Starlike & Error Convex Function,” Mathematics and Statistics, Vol.11, No.2, pp. 345-352, 2023. DOI: 10.13189/ms.2023.110214 (b): D Kavitha, K Dhanalakshmi, K Anitha, (2023). Toeplitz Determinant For Error Starlike & Error Convex Function. Mathematics and Statistics, 11(2), 345-352. DOI: 10.13189/ms.2023.110214 Copyright ©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Normalised Error function has been coined and analyzed in 2018 [13].The concept of normalised error function discussed in [13], motivated us to find the new results of Toeplitz determinant for the subclasses of analytic univalent functions concurrent with error function. By seeing the history of error function in Geometric functions theory, Ramachandran et. al [13] derived the coefficient estimates followed by the Fekete-Szeg¨ o problem for the normalised subclasses of starlike and convex functions associated with error function. Finding coefficient estimates is one of the most provoking concepts in geometric func- tion theory. In current scenario scientists are concentrating on special functions which are connected with univalent functions. Based on these concepts, the present paper deals with supremum and infimum of Toeplitz determinant for starlike and convex in terms of error function with convolution product using the concept of subordination. Also, we derive the sharp bounds for probability distribution associated with error starlike and error convex functions. Keywords Analytic Functions, Error Function, Toeplitz Determinant, Subordination and Fekete-Szeg¨ o Inequality 1 Introduction and preliminaries Many of the special function has been involved in applied mathematics and sciences. One of such function is called Error functions or Gaussian error function. It has a wide range of applications in the theory of optical sciences. A sigmoid-type function that occurs as a non-elementary function is treated as the error function er f . Error functions and numerical approximations can be applied in all domains of applied mathematics [5]. The Error function, which is also an entire function can be defined as, erf (z)= 2 √ π  z 0 exp(−u 2 )du = 2 √ π ∞  k=0 (−1) k z 2k+1 (2k + 1)k! . (1.1) In statistics, random variable can be derived as the error function. Whereas in complex analysis, error function will be analytic everywhere in regard to Taylor series. It has no singular points except from infinity. The Taylor expansion of error function is always convergent.