Application of generalized Bessel functions to classes of analytic functions
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Application of generalized Bessel functions to classes of analytic functions B. A. Frasin1 · Feras Yousef2 · Tariq Al-Hawary3 · Ibtisam Aldawish4 Received: 5 August 2020 / Accepted: 7 September 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract Let u p denote the normalized generalized Bessel function of the first kind of order p. The current work aims to determine a necessary and sufficient condition for the function z(2−u p (z)), the linear operator I (m, c) related to the normalized generalized Bessel function, z and the integral operator G (m, c, z) = 0 (2 − u p (t))dt to be in various subclasses of analytic functions. Some interesting special cases of our main results are considered. Keywords Analytic functions · Hadamard product · Generalized Bessel functions Mathematics Subject Classification 30C45 · 33C10
1 Introduction and definitions Bessel functions are needful in many branches of applied mathematics and mathematical physics, for example, those in acoustics, angular resolution, radio physics, hydrodynamics, and signal processing. Therefore, these special functions have been studied extensively, see [2,3,7,16,22]. Recently, there has been a great deal of interest on Bessel and hypergeometric functions from the point of view of geometric function theory, see [4,5,8,11,12,17].
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B. A. Frasin [email protected] Feras Yousef [email protected] Tariq Al-Hawary [email protected] Ibtisam Aldawish [email protected]
1
Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan
2
Department of Mathematics, The University of Jordan, Amman 11942, Jordan
3
Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan
4
Mathematics and Statistics Department, Imam Mohammed Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
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The generalized Bessel function of the first kind of order p, denoted by w p (see [2]), is defined as a particular solution of the second-order linear differential equation zw (z) + bzw (z) + cz 2 − p 2 + (1 − b) p w(z) = 0, where b, p, c ∈ C. It is known [2] that the analytic function w p is represented as the infinite series ∞ z 2n+ p (−1)n (c)n w p (z) = , z ∈ C. . b+1 n!( p + n + 2 ) 2 n=0
The normalized generalized Bessel function of the first kind of order p, denoted by u p , is defined with the transformation b + 1 − p/2 u p (z) = 2 p ( p + w p (z 1/2 ) )z 2 ∞ (−c/4)n n z , = (m)n n! n=0
where m = p + (b + 1)/2 = 0, −1, −2, . . . and (a)n is the well-known Pochhammer (or Appell) symbol, defined in terms of the Euler Gamma function for a = 0, −1, −2, . . . by (a + n) 1, if n = 0 (a)n = = a(a + 1)(a + 2) . . . (a + n − 1), if n ∈ N. (a) The function u p is analytic on C and satisfies the second-order linear differential equation 4z 2 u (z) + 2(2 p + b + 1)zu (z) + czu(z) = 0. For properties of the function u p , such as differentiation formula, integral representation, lower and upper
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