Bounds for the radii of univalence of some special functions

Abstract

Abstract. Tight lower and upper bounds for the radii of univalence (and starlikeness) of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions, and they are actually solutions of some functional equations. The Laguerre-Pólya class of entire functions plays an important role in our study.Mathematics subject classification (2010): 30C45, 30C15, 33C10.

Bounds for Radii of Starlikeness of Some $${\varvec{q}}$$ q -Bessel Functions

“…Also, the Laguerre-Pólya class LP of real entire functions, which consist of uniform limits of real polynomials whose zeros are all real, was used intensively (for more details on the Laguerre-Pólya class of entire functions we refer to [8] and to the references therein). Motivated by the earlier works, 1 ν is the smallest positive root of the equation r · dJ (2) ν (r; q)/dr = 0 and satisfies the following inequality…”

“…It is known that the Jackson and Hahn-Exton q-Bessel functions are q-extensions of the classical Bessel function of the first kind J ν . Clearly, for fixed z we have J (2) ν ((1 − z)q; q) → J ν (z) and J…”

“…The readers can find comprehensive information on the Bessel function of the first kind in Watson's treatise [25] and properties of Jackson and Hahn-Exton q-Bessel functions can be found in [18,19,21,22] and in the references therein. The geometric properties of some special functions (like Bessel, Struve and Lommel functions of the first kind) and their zeros in connection with these geometric properties were intensively studied by many authors (see [1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,20,23,26]). Also, the radii of starlikeness and convexity of some q-Bessel functions were investigated in [8].…”

“…is the smallest positive root of the equation r · dJ (2) ν (r; q)/dr − (ν − 1)J (2) ν (r; q) = 0 and satisfies the following inequalities…”

Geometric and monotonic properties of hyper-Bessel functions

“…Recently, there has been a vivid interest on some geometric properties such as univalency, starlikeness, convexity and uniform convexity of various special functions such as Bessel, Struve, Lommel, Wright and q-Bessel functions (see [1,2,3,4,7,8,9,10,11,15,16,13,17]). In the above mentioned papers the authors have used frequently some properties of the zeros of these functions.…”