On starlikeness of regular Coulomb wave functions. (English) Zbl 1515.30016
Summary: In this paper, we study some geometric properties of a class of analytic functions which is defined from the \(J\)-fraction expansion of the ratio \(zf'(z)/f(z)\). We find the disk domain which is mapped into a starlike domain by these functions. Moreover, we study similar results for two different normalized forms of regular Coulomb wave functions and a normalized Bessel function of the first kind by using continued fractions expansions.
MSC:
| 30B70 | Continued fractions; complex-analytic aspects |
| 11J70 | Continued fractions and generalizations |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 33C15 | Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) |
Software:
DLMFReferences:
| [1] | Akta\c{s}, \.{I}brahim, Geometric and monotonic properties of hyper-Bessel functions, Ramanujan J., 275-295 (2020) · Zbl 1481.30004 · doi:10.1007/s11139-018-0105-9 |
| [2] | Baricz, \'{A}rp\'{a}d, Radii of starlikeness of some special functions, Proc. Amer. Math. Soc., 3355-3367 (2016) · Zbl 1400.30017 · doi:10.1090/proc/13120 |
| [3] | Baricz, \'{A}rp\'{a}d, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc., 2019-2025 (2014) · Zbl 1291.30062 · doi:10.1090/S0002-9939-2014-11902-2 |
| [4] | Baricz, \'{A}rp\'{a}d, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct., 641-653 (2010) · Zbl 1205.30010 · doi:10.1080/10652460903516736 |
| [5] | Baricz, \'{A}rp\'{a}d, The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants, Linear Algebra Appl., 259-272 (2018) · Zbl 1390.15018 · doi:10.1016/j.laa.2018.03.012 |
| [6] | Baricz, \'{A}rp\'{a}d, Geometric properties of some Lommel and Struve functions, Ramanujan J., 325-346 (2017) · Zbl 1361.30021 · doi:10.1007/s11139-015-9724-6 |
| [7] | K. Bartschat, Computational Atomic Physics, Springer, Berlin, 1996. |
| [8] | Brown, R. K., Univalence of Bessel functions, Proc. Amer. Math. Soc., 278-283 (1960) · Zbl 0090.05003 · doi:10.2307/2032969 |
| [9] | Brown, R. K., Univalent solutions of \(W^{\prime \prime }, pW=0\), Canadian J. Math., 69-78 (1962) · Zbl 0108.08102 · doi:10.4153/CJM-1962-006-4 |
| [10] | Cuyt, Annie, Handbook of continued fractions for special functions, xvi+431 pp. (2008), Springer, New York · Zbl 1150.30003 |
| [11] | Hayden, T. L., Chain sequences and univalence, Illinois J. Math., 523-528 (1964) · Zbl 0122.07803 |
| [12] | H\"{a}st\"{o}, Peter, Starlikeness of the Gaussian hypergeometric functions, Complex Var. Elliptic Equ., 173-184 (2010) · Zbl 1187.30017 · doi:10.1080/17476930903276134 |
| [13] | Kreyszig, Erwin, The radius of univalence of Bessel functions. I, Illinois J. Math., 143-149 (1960) · Zbl 0091.06203 |
| [14] | K\"{u}stner, Reinhold, Mapping properties of hypergeometric functions and convolutions of starlike or convex functions of order \(\alpha \), Comput. Methods Funct. Theory, 597-610 (2002) · Zbl 1053.30006 · doi:10.1007/BF03321867 |
| [15] | Leighton, Walter, A general continued fraction expansion, Bull. Amer. Math. Soc., 596-605 (1939) · Zbl 0021.33004 · doi:10.1090/S0002-9904-1939-07046-8 |
| [16] | Merkes, E. P., On univalence of a continued fraction, Pacific J. Math., 1361-1369 (1960) · Zbl 0095.05803 |
| [17] | N. Michel, Precise Coulomb wave functions for a wide range of complex \(\ell , \eta\) and \(z\), Comput. Phys. Commun. 176 (2007) 232-249. · Zbl 1196.81030 |
| [18] | NIST handbook of mathematical functions, xvi+951 pp. (2010), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge · Zbl 1198.00002 |
| [19] | Ponnusamy, S., Univalence and convexity properties for confluent hypergeometric functions, Complex Variables Theory Appl., 73-97 (1998) · Zbl 0902.30011 · doi:10.1080/17476939808815101 |
| [20] | Ponnusamy, S., Univalence and convexity properties for Gaussian hypergeometric functions, Rocky Mountain J. Math., 327-353 (2001) · Zbl 0973.30017 · doi:10.1216/rmjm/1008959684 |
| [21] | Robertson, M. S., Schlicht solutions of \(W''+pW=0\), Trans. Amer. Math. Soc., 254-274 (1954) · Zbl 0057.31101 · doi:10.2307/1990768 |
| [22] | Ruscheweyh, St., On the order of starlikeness of hypergeometric functions, J. Math. Anal. Appl., 1-11 (1986) · Zbl 0598.30021 · doi:10.1016/0022-247X(86)90329-X |
| [23] | Scott, W. T., The corresponding continued fraction of a \(J\)-fraction, Ann. of Math. (2), 56-67 (1950) · Zbl 0035.33201 · doi:10.2307/1969497 |
| [24] | Wall, H. S., Analytic Theory of Continued Fractions, xiii+433 pp. (1948), D. Van Nostrand Co., Inc., New York, N. Y. · Zbl 0035.03601 |
| [25] | Wang, Li-Mei, On the order of convexity for the shifted hypergeometric functions, Comput. Methods Funct. Theory, 505-522 (2021) · Zbl 1476.30080 · doi:10.1007/s40315-021-00383-8 |
| [26] | Wilf, Herbert S., The radius of univalence of certain entire functions, Illinois J. Math., 242-244 (1962) · Zbl 0109.30203 |
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