대한수학회지 (Journal of the Korean Mathematical Society)

  • 제57권4호
  • /
  • Pages.1031-1052
  • /
  • 2020
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

RADII PROBLEMS FOR THE GENERALIZED MITTAG-LEFFLER FUNCTIONS

  • 투고 : 2019.08.20
  • 심사 : 2020.01.31
  • 발행 : 2020.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper our aim is to find various radii problems of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.

키워드

참고문헌

  1. 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
  2. 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
  3. I. Aktas, E. Toklu, and H. Orhan, Radii of uniform convexity of some special functions, Turkish J. Math. 42 (2018), no. 6, 3010-3024. https://doi.org/10.3906/mat-1806-43
  4. D. Bansal and J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ. 61 (2016), no. 3, 338-350. https://doi.org/10.1080/17476933.2015.1079628
  5. 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
  6. 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
  7. 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
  8. A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct. 21 (2010), no. 9, 641-653. https://doi.org/10.1080/10652460903516736
  9. A. Baricz and A. Prajapati, Radii of starlikeness and convexity of generalized Mittag-Leffler functions, Math. Commun. in press.
  10. 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
  11. A. Baricz, E. Toklu, and E. Kadioglu, Radii of starlikeness and convexity of Wright functions, Math. Commun. 23 (2018), no. 1, 97-117.
  12. 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
  13. R. Bharati, R. Parvatham, and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28 (1997), no. 1, 17-32. https://doi.org/10.5556/j.tkjm.28.1997.4330
  14. N. Bohra and V. Ravichandran, Radii problems for normalized Bessel functions of first kind, Comput. Methods Funct. Theory 18 (2018), no. 1, 99-123. https://doi.org/10.1007/s40315-017-0216-0
  15. D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, J. Lond. Math. Soc. s2-1 (1969), no. 1, 431-443. https://doi.org/10.1112/jlms/s2-1.1.431
  16. R. K. Brown, Univalence of Bessel functions, Proc. Amer. Math. Soc. 11 (1960), 278-283. https://doi.org/10.2307/2032969
  17. M. Caglar, E. Deniz, and R. Szasz, Radii of $\alpha$-convexity of some normalized Bessel functions of the first kind, Results Math. 72 (2017), no. 4, 2023-2035. https://doi.org/10.1007/s00025-017-0738-9
  18. E. Deniz and R. Szasz, The radius of uniform convexity of Bessel functions, J. Math. Anal. Appl. 453 (2017), no. 1, 572-588. https://doi.org/10.1016/j.jmaa.2017.03.079
  19. P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  20. M. M. Dzhrbashyan, Integral transforms and representations of functions in the complex domain, (in Russian), Nauka, Moscow, 1966.
  21. A. Gangadharan, V. Ravichandran, and T. N. Shanmugam, Radii of convexity and strong starlikeness for some classes of analytic functions, J. Math. Anal. Appl. 211 (1997), no. 1, 301-313. https://doi.org/10.1006/jmaa.1997.5463
  22. A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), no. 1, 87-92. https://doi.org/10.4064/ap-56-1-87-92
  23. S. Kanas and A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105 (1999), no. 1-2, 327-336. https://doi.org/10.1016/S0377-0427(99)00018-7
  24. S. Kanas and A. Wisniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl. 45 (2000), no. 4, 647-657.
  25. S. Kanas and T. Yaguchi, Subclasses of k-uniformly convex and starlike functions defined by generalized derivative. I, Indian J. Pure Appl. Math. 32 (2001), no. 9, 1275-1282.
  26. E. Kreyszig and J. Todd, The radius of univalence of Bessel functions. I, Illinois J. Math. 4 (1960), no. 1, 143-149. http://projecteuclid.org/euclid.ijm/1255455740 https://doi.org/10.1215/ijm/1255455740
  27. H. Kumar and A. M. Pathan, On the distribution of non-zero zeros of generalized Mittag-Leffler functions, Int. J. Eng. Res. Appl. 6, (2016), no.10, 66-71.
  28. E. P. Merkes and W. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc. 12 (1961), 885-888. https://doi.org/10.2307/2034382
  29. G. M. Mittag Leffler, Sur la nouvelle fonction $E_{\alpha}(x)$, C. R. Acad. Sci. Paris 137 (1903), 554-558.
  30. P. T. Mocanu, Une propriete de convexite generalisee dans la theorie de la representation conforme, Mathematica (Cluj) 11(34) (1969), 127-133.
  31. P. T. Mocanu and M. O. Reade, The radius of $\alpha$-convexity for the class of starlike univalent functions, $\alpha$-real, Proc. Amer. Math. Soc. 51 (1975), no. 2, 395-400. https://doi.org/10.2307/2040329
  32. I. V. Ostrovskii and I. N. Peresyolkova, Nonasymptotic results on distribution of zeros of the function $E_{\rho}(z,{\mu})$, Anal. Math. 23 (1997), no. 4, 283-296. https://doi.org/10.1007/BF02789843
  33. I. N. Peresyolkova, On distribution of zeros of generalized functions of Mittag-Leffler's type, Mat. Stud. 13 (2000), no. 2, 157-164.
  34. A. Yu. Popov and A. M. Sedletskii, Distribution of roots of Mittag-Leffler functions, J. Math. Sci. (N.Y.) 190 (2013), no. 2, 209-409; translated from Sovrem. Mat. Fundam. Napravl. 40 (2011), 3-171. https://doi.org/10.1007/s10958-013-1255-3
  35. T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15.
  36. J. K. Prajapat, S. Maharana, and D. Bansal, Radius of starlikeness and Hardy space of Mittag-Leffler functions, Filomat 32 (2018), no. 18, 6475-6486. https://doi.org/10.2298/FIL1818475P
  37. V. Ravichandran, On uniformly convex functions, Ganita 53 (2002), no. 2, 117-124.
  38. F. Ronning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), no. 1, 189-196. https://doi.org/10.2307/2160026
  39. S. Topkaya, E. Deniz, and M. Caglar, Radii of the $\beta$ uniformly convex of order $\alpha$ of Lommel and Struve functions, arXiv:1710.09715v2 [math.CV]
  40. H. S. Wilf, The radius of univalence of certain entire functions, Illinois J. Math. 6 (1962), no. 2, 242-244. http://projecteuclid.org/euclid.ijm/1255632321 https://doi.org/10.1215/ijm/1255632321
  41. A. Wiman, Uber die Nullstellen der Funktionen $E^a(x)$, Acta Math. 29 (1905), no. 1, 217-234. https://doi.org/10.1007/BF02403204