About the radius of starlikeness of Bessel functions of the first kind

Certain geometric properties of the Wright function

“…[1] Recently, several researchers studied classes of analytic functions involving special functions F ⊂ A, to find different conditions such that the members of F have certain geometric properties such as univalency, starlikeness or convexity in U. In this context many results are available in the literature regarding the generalized hypergeometric functions (see [2,3]) and Bessel functions (see [4][5][6][7][8][9][10]). In this paper, we study geometric properties of the Wright function, which is related to the Bessel function.…”

Close-to-Convexity of Some Special Functions and Their Derivatives

“…However, many important problems of Bessel functions, like determining the radius of starlikeness, and the radius of convexity, or finding the optimal parameter for which the normalized Bessel function of the first kind will be starlike, convex, or close-to-convex, have not been studied in details or have not been solved completely. Some of these problems have been studied later in the papers [1,2,3,5,6,8,19,20], however, there are still some open problems in this direction. For example, there is no information about the close-to-convexity or univalence of the derivatives of Bessel functions, or other special functions.…”

“…We note that the results of Theorem 1 are sharp. Moreover, it is worth to mention that Szász [19,Theorem 6] deduced already the starlikeness of f ν , while Baricz and Szász [8,Theorem 6] deduced already the convexity of f ν , however, our approach is much easier and as we can see below is applicable also for Struve and Lommel functions. Moreover, in the above theorems we have also information on the closeto-convexity or convexity of the derivatives of the Bessel, Struve and Lommel functions, respectively.…”

“…These problems remain open and are subject of further research. As we can see in the case of the Bessel functions of the first kind [5,8,19] these kind of problems are not easy to handle, since require a lot of information about the zeros of Bessel functions. But, the zeros of Struve and Lommel functions are not much studied; for example, there is no formula yet for their derivative with respect to the order, which would be an useful source in the study of the geometric properties of Struve and Lommel functions.…”

The Radius of $$\alpha $$ α -Convexity of Normalized Bessel Functions of the First Kind

“…We also mention that the univalence, starlikeness and convexity of Bessel functions of the first kind were studied extensively in several papers. We refer to [1][2][3][4][5][6][7][8][9]11,15,16] and to the references therein.…”