### Abstract

In this paper, we determine the radius of convexity for three kinds of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.

Original language | English |
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Journal | Analysis and Applications |

Issue number | 2 |

DOIs | |

Publication status | Accepted/In press - 2014 |

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### Keywords

- convex functions
- Dini function
- minimum principle for harmonic functions
- Normalized Bessel functions of the first kind
- radius of convexity
- residue theorem
- zeros of Bessel functions

### ASJC Scopus subject areas

- Applied Mathematics
- Analysis

### Cite this

*Analysis and Applications*, (2). https://doi.org/10.1142/S0219530514500316

**The radius of convexity of normalized Bessel functions of the first kind.** / Baricz, A.; Szász, Róbert.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The radius of convexity of normalized Bessel functions of the first kind

AU - Baricz, A.

AU - Szász, Róbert

PY - 2014

Y1 - 2014

N2 - In this paper, we determine the radius of convexity for three kinds of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.

AB - In this paper, we determine the radius of convexity for three kinds of normalized Bessel functions of the first kind. In the mentioned cases the normalized Bessel functions are starlike-univalent and convex-univalent, respectively, on the determined disks. The key tools in the proofs of the main results are some new Mittag-Leffler expansions for quotients of Bessel functions of the first kind, special properties of the zeros of Bessel functions of the first kind and their derivative, and the fact that the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind. Moreover, we find the optimal parameters for which these normalized Bessel functions are convex in the open unit disk. In addition, we disprove a conjecture of Baricz and Ponnusamy concerning the convexity of the Bessel function of the first kind.

KW - convex functions

KW - Dini function

KW - minimum principle for harmonic functions

KW - Normalized Bessel functions of the first kind

KW - radius of convexity

KW - residue theorem

KW - zeros of Bessel functions

UR - http://www.scopus.com/inward/record.url?scp=84902024489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902024489&partnerID=8YFLogxK

U2 - 10.1142/S0219530514500316

DO - 10.1142/S0219530514500316

M3 - Article

AN - SCOPUS:84902024489

JO - Analysis and Applications

JF - Analysis and Applications

SN - 0219-5305

IS - 2

ER -