Geometric properties of normalized Bessel, Struve and Lommel functions

In this thesis, lower and upper bounds for the radii of univalence, starlikeness and convexity of the some normalized Bessel, Struve and Lommel functions of the first kind are obtained by using the Euler-Rayleigh inequalities and some properties of the Laguerre-Pólya class of real entire functions. In addition, it is also shown that these radii are the smallest positive roots of some transcendental functional equations, and the radii of univalence of the some normalized Bessel, Struve and Lommel functions are exactly the radii of starlikeness of the same functions. On the other hand, the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, some new lower and upper bounds for the zeros of the derivatives of Struve and Lommel functions are obtained. The results on zeros of special functions may be of independent interest and can be useful in problems of mathematical physics where these zeros appear.