대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제57권2호
  • /
  • Pages.355-369
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  • 2020
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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BOUNDS FOR RADII OF CONVEXITY OF SOME q-BESSEL FUNCTIONS

  • Aktas, Ibrahim (Department of Mathematics Kamil Ozdag Science Faculty Karamanoglu Mehmetbey University) ;
  • Orhan, Halit (Department of Mathematics Faculty of Science Ataturk University)
  • 투고 : 2019.03.04
  • 심사 : 2019.08.19
  • 발행 : 2020.03.31
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초록

In the present investigation, by applying two different normalizations of the Jackson's second and third q-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-Pólya class of real entire functions plays an important role in this work.

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참고문헌

  1. I. Aktas and A. Baricz, Bounds for radii of starlikeness of some q-Bessel functions, Results Math. 72 (2017), no. 1-2, 947-963. https://doi.org/10.1007/s00025-017-0668-6
  2. I. Aktas, A. Baricz, and H. Orhan, Bounds for radii of starlikeness and convexity of some special functions, Turkish J. Math. 42 (2018), no. 1, 211-226. https://doi.org/10.3906/mat-1610-41
  3. I. Aktas, A. Baricz, and N. Yagmur, Bounds for the radii of univalence of some special functions, Math. Inequal. Appl. 20 (2017), no. 3, 825-843. https://doi.org/10.7153/mia-20-52
  4. A. Baricz, Geometric properties of generalized Bessel functions of complex order, Mathematica 48(71) (2006), no. 1, 13-18.
  5. A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), no. 1-2, 155-178.
  6. A. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, 1994, Springer-Verlag, Berlin, 2010. https://doi.org/10.1007/978-3-642-12230-9
  7. A. Baricz, D. K. Dimitrov, and I. Mezo, Radii of starlikeness and convexity of some q-Bessel functions, J. Math. Anal. Appl. 435 (2016), no. 1, 968-985. https://doi.org/ 10.1016/j.jmaa.2015.10.065
  8. A. Baricz, D. K. Dimitrov, H. Orhan, and N. Yagmur, Radii of starlikeness of some special functions, Proc. Amer. Math. Soc. 144 (2016), no. 8, 3355-3367. https://doi.org/10.1090/proc/13120
  9. A. Baricz, P. A. Kupan, and R. Szasz, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc. 142 (2014), no. 6, 2019-2025. https://doi.org/10.1090/S0002-9939-2014-11902-2
  10. A. Baricz, H. Orhan, and R. Szasz, The radius of $-\alpha}$-convexity of normalized Bessel functions of the first kind, Comput. Methods Funct. Theory 16 (2016), no. 1, 93-103. https://doi.org/10.1007/s40315-015-0123-1
  11. A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct. 21 (2010), no. 9-10, 641-653. https://doi.org/10.1080/10652460903516736
  12. A. Baricz and R. Szasz, The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl. (Singap.) 12 (2014), no. 5, 485-509. https://doi.org/10.1142/S0219530514500316
  13. A. Baricz and R. Szasz, Close-to-convexity of some special functions and their derivatives, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 1, 427-437. https://doi.org/10.1007/s40840-015-0180-7
  14. A. Baricz, E. Toklu, and E. Kadioglu, Radii of starlikeness and convexity of Wright functions, Math. Commun. 23 (2018), no. 1, 97-117.
  15. A. Baricz and N. Yagmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. 42 (2017), no. 2, 325-346. https://doi.org/10.1007/s11139-015-9724-6
  16. R. K. Brown, Univalence of Bessel functions, Proc. Amer. Math. Soc. 11 (1960), 278- 283. https://doi.org/10.2307/2032969
  17. P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  18. M. E. H. Ismail, The zeros of basic Bessel functions, the functions $J_{v+ax}$(x), and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1-19. https://doi.org/10.1016/0022-247X(82)90248-7
  19. M. E. H. Ismail and M. E. Muldoon, On the variation with respect to a parameter of zeros of Bessel and q-Bessel functions, J. Math. Anal. Appl. 135 (1988), no. 1, 187-207. https://doi.org/10.1016/0022-247X(88)90148-5
  20. M. E. H. Ismail and M. E. Muldoon, Bounds for the small real and purely imaginary zeros of Bessel and relatedfunctions, Methods Appl. Anal. 2 (1995), no. 1, 1-21. https://doi.org/10.4310/MAA.1995.v2.n1.a1
  21. H. T. Koelink and R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel functionand associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), no. 3, 690-710.https://doi.org/10.1006/jmaa.1994.1327
  22. T. H. Koornwinder and R. F. Swarttouw, On q-analogues of the Fourier and Hankeltransforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445-461. https://doi.org/10.2307/2154118
  23. E. Kreyszig and J. Todd, The radius of univalence of Bessel functions. I, Illinois J.Math. 4 (1960), 143-149. http://projecteuclid.org/euclid.ijm/1255455740
  24. B. Ya. Levin, Lectures on entire functions, translated from the Russian manuscript byTkachenko, Translations of Mathematical Monographs, 150, American MathematicalSociety, Providence, RI, 1996.
  25. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge UniversityPress, Cambridge, England, 1944.
  26. H. S. Wilf, The radius of univalence of certain entire functions, Illinois J. Math. 6(1962), 242-244. http://projecteuclid.org/euclid.ijm/1255632321 https://doi.org/10.1215/ijm/1255632321