Radii of starlikeness and convexity of some q-Bessel functions

A. Baricz, Dimitar K. Dimitrov, István Mezo

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 languageEnglish
Pages (from-to)968-985
Number of pages18
JournalJournal of Mathematical Analysis and Applications
Volume435
Issue number1
DOIs
Publication statusPublished - Mar 1 2016

Fingerprint

Starlikeness
Bessel functions
Bessel Functions
Convexity
Radius
Hyperbolic Polynomial
Bessel function of the first kind
Derivative
Interlacing
Derivatives
Entire Function
Argand diagram
Unit Disk
Normalization
Deduce
Open Problems
Factorization
Necessary Conditions
Sufficient Conditions
Polynomials

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

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

In: Journal of Mathematical Analysis and Applications, Vol. 435, No. 1, 01.03.2016, p. 968-985.

Research output: Contribution to journalArticle

Baricz, A. ; Dimitrov, Dimitar K. ; Mezo, István. / Radii of starlikeness and convexity of some q-Bessel functions. In: Journal of Mathematical Analysis and Applications. 2016 ; Vol. 435, No. 1. pp. 968-985.
@article{2a122c7b90f04a8c8775ea1575083bfc,
title = "Radii of starlikeness and convexity of some q-Bessel functions",
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-P{\'o}lya 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.",
keywords = "Interlacing property of zeros, Laguerre inequality, Laguerre-P{\'o}lya class of entire functions, Q-Bessel functions, Radius of starlikeness and convexity, Zeros of q-Bessel functions",
author = "A. Baricz and Dimitrov, {Dimitar K.} and Istv{\'a}n Mezo",
year = "2016",
month = "3",
day = "1",
doi = "10.1016/j.jmaa.2015.10.065",
language = "English",
volume = "435",
pages = "968--985",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "1",

}

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 -