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In this note our aim is to determine the radius of starlikeness of the normalized Bessel functions of the first kind for three different kinds of normalization. The key tool in the proof of our main result is the Mittag-Leffler expansion for Bessel functions of the first kind and the fact that, according to Ismail and Muldoon, the smallest positive zeros of some Dini functions are less than the first positive zero of the Bessel function of the first kind.

The integrals of the first type of double Bessel functions have a wide range of applications in geological exploration, mechanical and electromagnetic responses, signal processing, scattering, and wetting. In this paper, we develop a linear transformation accelerated convergence algorithm (LTACA) that combines the large argument approximate expression of the Bessel function (LAAEBF) and the integral accumulation to provide an efficient numerical algorithm for abnormal integrals with arbitrary order double Bessel functions of the first type. The effectiveness and high efficiency of the algorithm are verified by numerical examples, and its high accuracy is demonstrated by comparison with the Gaver-Stehfest inverse Laplace transform method (GSILTM). This offers a reliable and efficient computational method for the study of signal processing and mechanical problems.

In this paper we obtain the necessary and sufficient conditions for generalized Bessel functions of the first kind zup(z) to be in the classes S(b, λ, β) and R(b, λ, β) of analytic functions with complex order and also give the necessary and sufficient conditions for z(2−up(z)) to be in the classes TS(b, λ, β) and TR(b, λ, β). Furthermore, we give the necessary and sufficient conditions for J(k, c) to be in the class TR(b, λ, β) provided that the function f is in the class Rτ (A,B). Finally, we give conditions for the integral operator G(k, c, z) = z R0 (2 - up(t))dt to be in the class TR(b, λ, β). Several corollaries and consequences of the main results are also obtained.

In this paper, we prove monotonicity properties for the four different kinds of normalized q -Struve-Bessel functions using the method of subordination factor sequences. In addition, several inequalities related to the q -gamma function have been established. To support the main fundings, graphs derived from specific parameter values were presented. Additionality, in the special case of q → 1 , we obtain the results of Deniz and Szász (Complex Anal. Oper. Theory 18:120, 2004 ).

As a powerful tool for models of quantum computing, q-calculus has drawn the attention of many researchers in the discipline of special functions. In this paper, we present new properties and characterize q-Bessel functions of the first kind using some identities of q-calculus. The results presented in this article help us to obtain new expression results related to q-special functions. New summation and integral representations for q-Bessel functions of the first kind are also established. A few examples are also provided to demonstrate the effectiveness of the proposed strategy.

This study explores the geometric properties of normalized Gaussian hypergeometric functions in a certain subclass of analytic functions. This work investigates the inclusion properties of integral operators associated with generalized Bessel functions of the first kind. By finding the conditions on the parameters of the Gaussian hypergeometric function, we identify criteria for the normalized Gaussian hypergeometric function to be in a certain subclass of analytic functions. Using Taylor coefficients, we obtained sufficient conditions for the integral operators associated with generalized Bessel functions of the first kind to belong to different subclasses of univalent functions. The results obtained are analyzed and compared with the existing literature, providing new insights into the subclasses.

In this paper, we study certain geometric properties such as the starlikeness of order ζ and the convexity of order ζ of the generalized k-Bessel function. In addition, we establish several requirements for the parameters so that the generalized k-Bessel function belongs to some subclasses of analytic functions. Furthermore, as an application of the geometric properties, we establish certain results concerning the Hardy spaces of the generalized k-Bessel function. On the other hand, we present some corollaries concerning the classical Bessel function Jρ and the modified Bessel function Iρ. To support our geometric results, we present some specific examples of functions that map the open unit disk onto the symmetric domains with respect to the real axis.

In this study, we extend and refine several results concerning the geometric properties of generalized Bessel functions established by Á. Baricz (Mathematica 48(71):1318, 2006 ). The analysis focuses on cases where the parameters lie within a bounded domain. The primary methodology employs classical sufficient conditions for univalency, convexity, starlikeness, and close-to-convexity of analytic functions defined within the open unit disk, as originally formulated by Ozaki (Sci. Rep. Tokyo Bunrika Daigaku 2:167–188, 1935 ) and Mocanu (Libertas Math. 13:27–40, 1993 ). Additionally, this work explores the uniform convexity and starlikeness of the normalized form of the Bessel function. To demonstrate the validity and applicability of the theoretical framework, several specific examples are provided.

There are many articles in the literature dealing with the first-order and the second-order differential subordination and differential superordination problems for analytic functions in the unit disk, but there are only a few articles dealing with the third-order differential subordination problems. The concept of third-order differential subordination in the unit disk was introduced by Antonino and Miller, and studied recently by Tang and Deniz. Let Ω be a set in the complex plane C, let p(z) be analytic in the unit disk U={z:z∈Cand|z|<1}, and let ψ:C4×U→C. In this paper, we investigate the problem of determining properties of functions p(z) that satisfy the following third-order differential superordination: Ω⊂ψ(p(z),zp′(z),z2p′′(z),z3p′′′(z);z):z∈U.As applications, we derive some third-order differential superordination results for analytic functions in U, which are associated with a family of generalized Bessel functions. The results are obtained by considering suitable classes of admissible functions. New third-order differential sandwich-type results are also obtained.

A method for the approximate solution of the scattering problem for the Zakharov-Shabat system with a finitely supported potential is presented, based on the construction of the Delsarte transmutation operators. For the solutions of the Zakharov-Shabat system functional series representations are obtained in the form of the Neumann series of Bessel functions. The representations lend themselves for the numerical computation.