symmetry S S Article On Convex Functions Associated with Symmetric Cardioid Domain Sarfraz Nawaz Malik 1 , Mohsan Raza 2 , Qin Xin 3 , Janusz Sokól 4 , Rabbiya Manzoor 1 and Saira Zainab 5, *   Citation: Malik, S.N.; Raza, M.; Xin, Q.; Sokól, J.; Manzoor, R.; Zainab, S. On Convex Functions Associated with Symmetric Cardioid Domain. Symmetry 2021, 13, 2321. https:// doi.org/10.3390/sym13122321 Academic Editors: Derek Thomas, Nak Eun Cho and Ioan Ras , a Received: 10 November 2021 Accepted: 26 November 2021 Published: 4 December 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, Wah Campus, COMSATS University Islamabad, Wah Cantt 47040, Pakistan; snmalik110@ciitwah.edu.pk or snmalik110@yahoo.com (S.N.M.); ra.manzoor@yahoo.com (R.M.) 2 Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan; mohsanraza@gcuf.edu.pk or mohsan976@yahoo.com 3 Faculty of Science and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands; qinx@setur.fo 4 Department of Mathematics, Rzeszów University of Technology, Al. Powstañców Warszawy 12, 35-959 Rzeszów, Poland; jsokol@prz.rzeszow.pl 5 School of Electrical Engineering and Computer Science (SEECS), National University of Sciences & Technology (NUST), Sector H-12, Islamabad 44000, Pakistan * Correspondence: saira.zainab@seecs.edu.pk Abstract: The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, new geometrical structures are introduced and studied as an image domain and then their subsequent analytic functions are defined. Inspired and motivated by ongoing research, Malik et al. introduced a very innovative domain named the cardioid domain, which is symmetric about a real axis. Extending the same work on this symmetric cardioid domain, in this article, we provide a deeper analysis and define and study the convex functions associated with the symmetric cardioid domain, named cardio-convex functions. Keywords: analytic functions; shell-like curve; Fibonacci numbers; symmetric cardioid domain; convex functions; cardio-convex functions 1. Introduction In classical mathematics, the theory of analytic functions is one of the outstanding and elegant parts. In this theory, we study the analytic structure as well as the geometric properties of univalent and multivalent functions. In recent decades, there has been remarkable growth in the research on structural and geometrical properties of analytic functions. We can see the applications of analytic functions in mathematics such as in complex analysis, algebraic geometry, and number theory. Other than mathematical analysis, these functions are extensively used in various fields including fractional calculus, ODEs and PDEs, and operators’ theory, to name a few. There are many other problems in physics and other sciences that use differential equations and benefit from analytic functions. An interesting fact is that the relationship between the theory of analytic function and the logarithmic potential is the same as that between the theory of three-dimensional functions and the Newtonian potential. Moreover, some results of potential theory can be studied in the framework of this theory. Analytic functions have also been used in image processing to define the mathematical background of analytic signals. In the late 40s, in the framework of communication theory, analytical signals were introduced. From then onward, these signals were used to represent real valued signals. To give an idea of the approach taken to relate the analytic functions and analytic signals, one can notice that the analytic signals represent the boundary values of an analytic function in the upper half plane or that the periodic signals represent the Symmetry 2021, 13, 2321. https://doi.org/10.3390/sym13122321 https://www.mdpi.com/journal/symmetry