Certain geometric properties of two variable generalized Bessel functions

Volume 40, Issue 1, pp 49--72 https://dx.doi.org/10.22436/jmcs.040.01.04

Publication Date: May 30, 2025 Submission Date: March 10, 2025 Revision Date: March 29, 2025 Accteptance Date: April 04, 2025

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Authors

S. A. H. Shah - Department of Mathematics, University of Sargodha, Sargodha, Pakistan. Hafsa - Department of Mathematics, University of Sargodha, Sargodha, Pakistan. F. A. Awwad - Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia. E. A. A. Ismail - Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia. T. Gul - Cambridge Graphene Center, University of Cambridge, 9 JJ Thomson Ave, Cambridge, CB3 0FA, UK. - Department of Mathematics, City University of Science and Information Technology Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan.

Abstract

In this paper, we will define the normalized form of the generalized Bessel functions in \(k\) and \(s,k\) form. Sufficient conditions will be given under this study for starlikeness and convexity of normalized forms of \(k\)-Bessel function and \(s,k\)-Bessel function. Geometrical interpretation of Generalized Bessel \(k\) and \(s,k\) function for different values of \(k\) and \(s\) will also be discussed. For better understanding of the reader, some examples will be provided regarding to our approach. The graphical behaviour of these normalized functions and a comparison of graphs with classical form will also be studied to show the accuracy of results.

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Keywords

• Analytic
• univalent
• starlikeness
• convexity
• Pochhammer
• Gamma function
• generalized Bessel function

MSC

•  30C45
•  33C10
•  33B15

References

• [1] R. S. Ali, S. Mubeen, K. S. Nisar, S. Araci, G. Rahman, Some properties of generalized (s, k)-Bessel function in two variables, J. Math. Comput. Sci., 24 (2020), 10–21
  • View Article
  • Google Scholar


• [2] F. M. Arscott, The land beyond Bessel: A survey of higher special functions, In: Ordinary and partial differential equations (Proc. Sixth Conf., Univ. Dundee, Dundee, 1980), Springer, Berlin, 846 (1981), 26–45
  • View Article
  • Google Scholar
  • MATH


• [3] M. A. Boston, Birkhäuser, National Institute of Standards, Digital Library of Mathematical Functions, (1988)
  • View Article


• [4] T. Bulboac˘a, H. M. Zayed, Analytical and geometrical approach to the generalized Bessel function, J. Inequal. Appl., 2024 (2024), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] C. Cesarano, D. Assante, A note on Generalized Bessel Functions, Int. J. Math. Models Methods Appl. Sci., 7 (2013), 625–629
  • View Article
  • Google Scholar


• [6] O. P. Dave, Special Functions in Mathematics, Int. J. Adv. Res. Electr., Electron. Instrum. Eng., 5 (2016), 7651–7655
  • View Article


• [7] R. Díaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol, Divulg. Mat., 15 (2007), 179–192
  • View Article
  • Google Scholar
  • MATH


• [8] B. A. Frasin, I. Aldawish, On subclasses of uniformly spiral-like functions associated with generalized Bessel functions, J. Funct. Spaces, 2019 (2019), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] D. Girela, Basic theory of univalent functions, In: Conference Paper, Complex Anal. Rel. Areas, (2013)
  • View Article


• [10] F. A. Idris, A. L. Buhari, T. U. Adamu, Bessel Functions and Their Applications: Solution to Schrödinger equation in a cylindrical function of the second kind and Hankel Functions, Int. J. Novel Res. Phys. Chem. Math., 3 (2016), 17–31
  • View Article
  • Google Scholar


• [11] V. Kiryakova, A guide to special functions in fractional calculus, Mathematics, 9 (2021), 40 pages
  • View Article
  • Google Scholar


• [12] B. S. Koranga, S. K. Padaliya, V. K. Nautiyal, Special Functions and Their Application, River Publishers, New York (2022)
  • View Article
  • Google Scholar


• [13] N. N. Lebedev, Special functions and their applications, Dover Publications, New York (1972)
  • View Article
  • Google Scholar


• [14] D. Liu, M. U. Din, M. Raza, S. N. Malik, H. Tang, Convexity, Starlikeness, and Prestarlikeness of Wright Functions, Mathematics, 10 (2022), 15 pages
  • View Article
  • Google Scholar


• [15] Y. L. Luke, The special functions and their approximations, Vol. I, Academic Press, New York-London (1969)
  • View Article


• [16] Y. L. Luke, The special functions and their approximations. Vol. II, Academic Press, New York-London (1969)
  • View Article
  • Google Scholar


• [17] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer-Verlag, New York (1966)
  • View Article
  • Google Scholar
  • MATH


• [18] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London (2010)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] S. Mubeen, R. S. Ali, I. Nayab, G. Rahman, T. Abdeljawad, K. S. Nisar, Integral transforms of an extended generalized multi-index Bessel function, AIMS Math., 5 (2020), 7531–7546
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] S. Mubeen, S. Shah, G. Rahman, K. Nisar, T. Abdeljawad, Some Generalized Special Functions and their Properties, Adv. Theory Nonlinear Anal. Appl., 6 (2022), 45–65
  • View Article
  • Google Scholar


• [21] T. Nahid, M. Ali, Several characterizations of Bessel functions and their applications, Georgian Math. J., 29 (2022), 83–93
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] F. J. Narcowich, Notes on Special Functions, Texas A&M University, (2005)
  • View Article
  • Google Scholar


• [23] J. Niedziela, Bessel Functions and Their Applications, University of Tennessee-Knoxville, (2008)
  • View Article
  • Google Scholar


• [24] A. F. Nikiforov, V. B. Uvarov, Special functions of mathematical physics, Birkhäuser Verlag, Basel (1988)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] K. Parand, M. Nikarya, Application of Bessel functions for solving differential and integro-differential equations of the fractional order, Appl. Math. Model., 38 (2014), 4137–4147
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Gordon and Breach Science Publishers, Australia (1992)
  • View Article
  • Google Scholar


• [27] R. Roy, Sources in the Development of Mathematics, Cambridge University Press, New York (2011)
  • View Article
  • Google Scholar


• [28] M. K. Sajid, R. S. Ali, I. Nayab, Some results of generalized k-fractional integral operator with k-Bessel function, Turk. J. Sci., 5 (2020), 157–169
  • View Article
  • Google Scholar


• [29] H. M. Srivastava, An Introductory Overview of Bessel Polynomials, the Generalized Bessel Polynomials and the q-Bessel Polynomials, Symmetry, 15 (2023), 28 pages
  • View Article
  • Google Scholar


• [30] D. L. Suthar, A. M. Khan, A. Alaria, S. D. Purohit, J. Singh, Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus, AIMS Math., 5 (2020), 1400–1410
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] N. M. Temme, Special functions: An introduction to the classical functions of mathematical physics, John Wiley & Sons, New York (2011)
  • View Article
  • Google Scholar


• [32] D. K. Thomas, N. Tuneski, A. Vasudevarao, Univalent functions, De Gruyter, Berlin (2018)
  • View Article
  • MathSciNet
  • Google Scholar


• [33] G. N. Watson, A treatise on the theory of Bessel functions, The University Press, Cambridge (1922)
  • View Article
  • Google Scholar


• [34] N. Yagmur, H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal., 2013 (2013), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] H. M. Zayed, T. Bulboac˘a, Normalized generalized Bessel function and its geometric properties, J. Inequal. Appl., 2022 (2022), 26 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH