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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aktaş, I., Baricz, Á., Yağmur, N.: Bounds for the radii of univalence of some special functions. Math. Inequal. Appl. 20(3), 825–843 (2017)
Al-Kharsani, H.A., Baricz, Á., Pogány, T.K.: Starlikeness of a cross-product of Bessel functions. J. Math. Inequal. 10(3), 819–827 (2016)
Ashbaugh, M.S., Benguria, R.D.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78(1), 1–17 (1995)
Baricz, Á., Dimitrov, D.K., Mező, I.: Radii of starlikeness and convexity of some \(q\)-Bessel functions. J. Math. Anal. Appl. 435, 968–985 (2016)
Baricz, Á., Kupán, P.A., Szász, R.: The radius of starlikeness of normalized Bessel functions of the first kind. Proc. Am. Math. Soc. 142(6), 2019–2025 (2014)
Baricz, Á., Ponnusamy, S., Singh, S.: Cross-product of Bessel functions: monotonicity patterns and functional inequalities. Proc. Indian Sci. (Math. Sci.) (in press)
Baricz, Á., Singh, S.: Zeros of some special entire functions. Proc. Am. Math. Soc. 146(5), 2207–2216 (2018)
Baricz, Á., Szász, R.: The radius of convexity of normalized Bessel functions of the first kind. Anal. Appl. 12(5), 485–509 (2014)
Baricz, Á., Yağmur, N.: Geometric properties of some Lommel and Struve functions. Ramanujan J. 42(2), 325–346 (2017)
Baricz, Á., Yağmur, N.: Radii of starlikeness and convexity of a cross-product of Bessel functions. Ramanujan J. 44(3), 493–519 (2017)
Brown, R.K.: Univalence of Bessel functions. Proc. Am. Math. Soc. 11(2), 278–283 (1960)
Brown, R.K.: Univalent solutions of \(W^{\prime \prime }+pW=0,\) Canad. J. Math. 14, 69–78 (1962)
Brown, R.K.: Univalence of normalized solutions of \(W^{\prime \prime }(z)+p(z)W(z)=0,\) Int. J. Math. Math. Sci. 5, 459–483 (1982)
Dimitrov, D.K., Ben Cheikh, Y.: Laguerre polynomials as Jensen polynomials of Laguerre–Pólya entire functions. J. Comput. Appl. Math. 233, 703–707 (2009)
Ismail, M.E.H., Muldoon, M.E.: Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2(1), 1–21 (1995)
Ki, H., Kim, Y.O.: On the number of nonreal zeros of real entire functions and the Fourier–Pólya conjecture. Duke Math. J. 104(1), 45–73 (2000)
Kreyszig, E., Todd, J.: The radius of univalence of Bessel functions. Ill. J. Math. 4, 143–149 (1960)
Lorch, L.: Monotonicity of the zeros of a cross-product of Bessel functions. Methods Appl. Anal. 1(1), 75–80 (1994)
Merkes, E.P., Robertson, M.S., Scott, W.T.: On products of starlike functions. Proc. Am. Math. Soc. 13, 960–964 (1962)
Nehari, Z.: The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Özer, S., Şengül, T.: Stability and transitions of the second grade Poiseuille flow. Phys. D 331, 71–80 (2016)
Robertson, M.S.: Schlicht solutions of \(W^{\prime \prime }+pW=0\). Trans. Am. Math. Soc. 76, 254–274 (1954)
Runckel, H.J.: Zeros of entire functions. Trans. Am. Math. Soc. 143, 343–362 (1969)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)
Wilf, H.S.: The radius of univalence of certain entire functions. Ill. J. Math. 6, 242–244 (1962)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Beril Kübra, Boróka and Koppány.
Rights and permissions
About this article
Cite this article
Baricz, Á., Szakál, A., Szász, R. et al. Radii of Starlikeness and Convexity of a Product and Cross-Product of Bessel Functions. Results Math 73, 62 (2018). https://doi.org/10.1007/s00025-018-0823-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0823-8