### Abstract

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler–Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.

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

Article number | 62 |

Journal | Results in Mathematics |

Volume | 73 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 2018 |

### Fingerprint

### Keywords

- Bessel and modified Bessel functions of the first kind
- distribution of zeros of entire functions
- fourier critical points
- Laguerre–Pólya class of entire functions
- radius of starlikeness and convexity
- zeros of hypergeometric polynomials

### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics

### Cite this

*Results in Mathematics*,

*73*(2), [62]. https://doi.org/10.1007/s00025-018-0823-8

**Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions.** / Baricz, A.; Szakál, Anikó; Szász, Róbert; Yağmur, Nihat.

Research output: Contribution to journal › Article

*Results in Mathematics*, vol. 73, no. 2, 62. https://doi.org/10.1007/s00025-018-0823-8

}

TY - JOUR

T1 - Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions

AU - Baricz, A.

AU - Szakál, Anikó

AU - Szász, Róbert

AU - Yağmur, Nihat

PY - 2018/6/1

Y1 - 2018/6/1

N2 - The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler–Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.

AB - The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler–Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.

KW - Bessel and modified Bessel functions of the first kind

KW - distribution of zeros of entire functions

KW - fourier critical points

KW - Laguerre–Pólya class of entire functions

KW - radius of starlikeness and convexity

KW - zeros of hypergeometric polynomials

UR - http://www.scopus.com/inward/record.url?scp=85045414695&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045414695&partnerID=8YFLogxK

U2 - 10.1007/s00025-018-0823-8

DO - 10.1007/s00025-018-0823-8

M3 - Article

AN - SCOPUS:85045414695

VL - 73

JO - Results in Mathematics

JF - Results in Mathematics

SN - 0378-6218

IS - 2

M1 - 62

ER -