The Radii of Starlikeness for Concave Functions

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