### Abstract

Geometric properties of the Jackson and Hahn-Exton q-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre-Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn-Exton q-Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q-Bessel function and its derivatives. Some open problems are proposed at the end of the paper.

Original language | English |
---|---|

Pages (from-to) | 968-985 |

Number of pages | 18 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 435 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 2016 |

### Fingerprint

### Keywords

- Interlacing property of zeros
- Laguerre inequality
- Laguerre-Pólya class of entire functions
- Q-Bessel functions
- Radius of starlikeness and convexity
- Zeros of q-Bessel functions

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*435*(1), 968-985. https://doi.org/10.1016/j.jmaa.2015.10.065

**Radii of starlikeness and convexity of some q-Bessel functions.** / Baricz, A.; Dimitrov, Dimitar K.; Mezo, István.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 435, no. 1, pp. 968-985. https://doi.org/10.1016/j.jmaa.2015.10.065

}

TY - JOUR

T1 - Radii of starlikeness and convexity of some q-Bessel functions

AU - Baricz, A.

AU - Dimitrov, Dimitar K.

AU - Mezo, István

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Geometric properties of the Jackson and Hahn-Exton q-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre-Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn-Exton q-Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q-Bessel function and its derivatives. Some open problems are proposed at the end of the paper.

AB - Geometric properties of the Jackson and Hahn-Exton q-Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q-generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre-Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn-Exton q-Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q-Bessel function and its derivatives. Some open problems are proposed at the end of the paper.

KW - Interlacing property of zeros

KW - Laguerre inequality

KW - Laguerre-Pólya class of entire functions

KW - Q-Bessel functions

KW - Radius of starlikeness and convexity

KW - Zeros of q-Bessel functions

UR - http://www.scopus.com/inward/record.url?scp=84960422791&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960422791&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2015.10.065

DO - 10.1016/j.jmaa.2015.10.065

M3 - Article

AN - SCOPUS:84960422791

VL - 435

SP - 968

EP - 985

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -