The Radii of Starlikeness for Concave Functions

Abstract

Let <img src=image/13426735_02.gif> denote the functions' class that is normalized, analytic, as well as univalent in the unit disc given by <img src=image/13426735_01.gif>. Convex, starlike, as well as close-to-convex functions resemble the main subclasses of <img src=image/13426735_02.gif>, expressed by <img src=image/13426735_03.gif>, as well as <img src=image/13426735_04.gif>, accordingly. Many mathematicians have recently studied radius problems for various classes of functions contained in <img src=image/13426735_02.gif>. The determination of the univalence radius, starlikeness, and convexity for specific special functions in <img src=image/13426735_02.gif> is a relatively new topic in geometric function theory. The problem of determining the radius has been initiated since the 1920s. Mathematicians are still very interested in this, particularly when it comes to certain special functions in <img src=image/13426735_02.gif>. Indeed, many papers investigate the radius of starlikeness for numerous functions. With respect to the open unit disc <img src=image/13426735_05.gif> and class <img src=image/13426735_02.gif>, the class of concave functions <img src=image/13426735_06.gif>, known as <img src=image/13426735_07.gif>, is defined. It is identified as a normalised analytic function <img src=image/13426735_08.gif>, which meets the requirement of having the opening angle of <img src=image/13426735_09.gif> at <img src=image/13426735_10.gif>. A univalent function <img src=image/13426735_11.gif> is known as concave provided that <img src=image/13426735_12.gif> is concave. In other words, we have that <img src=image/13426735_13.gif> is also convex. There is no literature to date on determining the radius of starlikeness for concave univalent functions related to certain rational functions, lune, cardioid, and the exponential equation. Hence, by employing the subordination method, we present new findings on determining several radii of starlikeness for different subclasses of starlike functions for the class of concave univalent functions <img src=image/13426735_07.gif>.