Certain geometric properties of the fractional integral of the Bessel function of the first kind

: This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order λ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and su ffi cient univalence conditions for the new fractional calculus operator using the methods associated to di ff erential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in di ff erential subordination and superordination theories in particular.


Introduction
Fractional calculus has recently become a prominent area of mathematical analysis with regard to theoretical studies and real-life applications.It has developed into an essential tool for modeling and analysis, having a significant impact on a wide range of disciplines.Recent detailed reviews of the subject [1,2] examine its development and list numerous scientific and engineering fields in which it has been applied.
The recent review study by Srivastava [3] highlighted the advantages of adding fractional calculus to geometric function theory, and this has inspired and encouraged further research linking fractional calculus to univalent functions theory.Since Miller and Mocanu first proposed the theory of differential subordination in their publications from 1978 [4] and 1981 [5], it has received a lot of attention because it can be used to more easily obtain previously known results as well as generate remarkable results when combined with investigations involving analytic functions.A topic of study that evolved well in the differential subordination theory framework emerged when various kinds of operators were included into the study.As a recent survey report [6] demonstrates, integral operators are a crucial tool when such studies are taken into account.
The investigation addressed in this paper continues this research topic and geometric properties are established for an integral operator introduced using the fractional integral of order λ and the remarkable Bessel function of the first kind.
The classical fractional calculus operators, Riemann-Liouville fractional integral of order λ and Riemann-Liouville fractional derivative of order λ, were adapted for studies regarding analytic functions in [7].The Riemann-Liouville fractional integral seen in [7] was combined with the generalized hypergeometric function in [8] and with the Gauss hypergeometric function in a general family of fractional integral operators that were introduced in [9].A unified method on special functions and fractional calculus operators was introduced in [10] including a list of publications that can be read to follow the topic's evolution.Studies involving fractional operators in geometric function theory continued over the years.In recent developments on the topic, the Riemann-Liouville fractional integral was combined with the confluent hypergeometric function [11], Gauss hypergeometric function [12], Libera integral operator [13], Mittag-Leffler-Confluent hypergeometric functions [14], or with the q-hypergeometric function [15].
The present research continues the direction of study regarding the development of new fractional operators for conducting studies in geometric function theory.The methods are similar to those used in the previously cited works, but the new results embedding fractional calculus aspects are developed here using the prolific methods of the differential subordination or the admissible functions theory [16] and a well-known result due to Sakaguchi.Such combination of methods has not been used earlier for research on fractional integral operators.
Moreover, the novelty of the research is completed by the combination of the Riemann-Liouville fractional integral of order λ with the form introduced in [7] and the Bessel function of the first kind, resulting in a new fractional integral operator that has not been investigated before.As proved earlier, the Riemann-Liouville fractional integral is widely used in theoretical studies pertaining new fractional operators involving different hypergeometric functions with applications in the geometric function theory.The new application of this prolific investigation tool on the Bessel function of the first kind presented in this study is motivated by the initial studies on its geometric properties established by Baricz Á. [17][18][19] and more recent studies seen in papers [20][21][22][23].
The significant results published recently regarding an operator defined using the Atangana-Baleanu fractional integral applied to Bessel functions [24] constituted a confirmation of the hypothesis that an interesting fractional operator would be obtained by combining the fractional integral of order λ given in [7] and the Bessel function of the first kind.Further investigations conducted in this paper on the fractional integral operator introduced here involving the Bessel function of the first kind were inspired by the intriguing results recently published regarding geometric properties of integral operators defined involving the Bessel function [25][26][27].
With the topic of the study exposed and the motivation shown above, we now introduce the basic notions and denotations familiar to the geometric function theory.
The general class of holomorphic functions in the unit disc U = {z ∈ C : |z| < 1} is identified by H(U).The notations are associated to the unit disc U.
The special classes with A 1 = A are also involved in this study.Furthermore, the study requires the following prominent classes: with α < 1, denoting the class of starlike functions of order α, obtaining the class of starlike functions S * when α = 0, and with α < 1, denoting the class of convex functions of order α, obtaining the class of convex functions K when α = 0.
Finally, the study also includes reference to the class of Carathéodory functions, and the class of close-to-convex functions, Such a function is called close-to-convex with respect to the identical function and the condition gives that a close-to-convex function is also univalent in U.
A known result, which will be used in the following proofs, is given as: Lemma 1.1.[28, p. 84] A necessary and sufficient condition for a function f ∈ H(U), f ′ (z) 0, to be close-to-convex is: The basic concept of the differential subordination is known as the following: Definition 1.1.[7,29] Let f and F be members of H(U).The function f is said to be subordinate to F, written f ≺ F, f (z) ≺ F(z), if there exists a function w analytic in U, with w(0) = 0 and |w(z The remarkable result due to Sakaguchi is a notable tool for obtaining the new outcome of this study. Lemma 1.2.[28, p. 49] Let p ∈ H(U), such that Re p(0) > 0, and let α ∈ R. Then The essential research tools are presented next.
Definition 1.2.[7,8] The fractional integral of order λ (λ > 0) is defined for a function f by the following expression: where f is an analytic function in a simply-connected region of the z-plane containing the origin and the multiplicity of (z − t) 1−λ is removed by requiring log(z − t) to be real when z − t > 0.
Definition 1.3.[17] Consider the second-order differential equation which is called Bessel's equation, where ν ∈ R or ν ∈ C and z ∈ C. The particular solutions of this equation are called Bessel functions.
The Bessel function of the first kind and order ν is given by: where Γ is Euler's gamma function.
The key findings of this investigation are revealed in the next section, starting with the introduction of the new fractional integral operator defined that considers Definitions 1.2 and 1.3.The first new results established in this paper give necessary and sufficient conditions for the newly defined operator to be starlike of positive and negative order, respectively.Next, similar conditions are obtained such that the newly defined operator is convex of positive and negative order, respectively.Furthermore, it is proved that the convexity of the fractional integral of the Bessel function of the first kind and order 0 implies its starlikeness.Finally, the conditions for the fractional integral of the Bessel function of the first kind and order 0 to be a close-to-convex function are obtained.

Results
The first new result introduced by this research shows the integral operator given as the fractional integral of the Bessel function of the first kind and order ν ≥ 0. The definition is obtained using Definitions 1.2 and 1.3 and the properties of the functions involved.Definition 2.1.Let ν ≥ 0, z ∈ C, λ > 0, p ∈ N, the fractional integral of order λ be given by (1.1), and the Bessel function of the first kind and order ν both be given by (1.2).The fractional integral of the Bessel function of the first kind and order ν are defined as: Remark 2.1.For ν = 0, we obtain: Furthermore, for λ = 1, we get the following particular form, which is important for this investigation; the fractional integral of the Bessel function of the first kind and order 0: 3) The following theorems highlight the necessary and sufficient conditions for the operator D −λ z I ν (z) introduced in (2.1) to be starlike and starlike of positive and negative order, respectively.Theorem 2.1.Let D −λ z I ν (z) be given by (2.1) and consider q ∈ K as: Proof.Since q ∈ K, q(0) = 1, and We can only conclude that D −λ z I ν (z) ∈ P. For λ = 1, ν = 0, and considering Relation (2.3), we have that (D −1 z I 0 ) ′ (0) = 1, and the following corollary is obtained: z I 0 (z) as given by (2.3) and consider q ∈ K as: ) , we have that g(0) = 1, and by using (2.6), we can write Re g(z) > 0, z ∈ U; hence, g(z) ∈ P.
The following results demonstrate the necessary and sufficient conditions for the fractional integral of the Bessel function of the first kind and order ν given by (2.1) to be starlike of positive order.Theorem 2.2.Let D −λ z I ν (z) as given by (2.1) and consider q ∈ K as: Proof.First, it is proved that q(z) = z 1 + z ∈ K.For that, we get Since q(0) = 0 and q ′ (0) = 1 0, Relation (2.8) gives that q(z) ∈ K.It follows that is a convex domain and Relation (2.7) is equivalent to: (2.9) For λ = 1, ν = 0, and considering Relation (2.3), we have that (D −1 z I 0 ) ′ (0) = 1, and the following corollary is obtained: z I 0 (z) as given by (2.3) and consider q ∈ K as: Proof.Since (D −1 z I 0 ) ′ (0) = 1 0, Inequality (2.10) shows that D −1 z I 0 (z) ∈ S * 1 2 .We mentioned in Remark 1.1 from the Introduction that for α ∈ [0, 1), we have that S * 1 2 ⊂ S * ; hence, D −1 z I 0 (z) ∈ S . □ The next results demonstrate the necessary and sufficient conditions for the fractional integral of the Bessel function of the first kind and order ν given by (2.1) to be starlike of negative order.Theorem 2.3.Let D −λ z I ν (z) as given by (2.1) and consider q ∈ K as: then Proof.The proof begins by proving that q(z) = z 1 − z ∈ K.For that, we have Since q ′ (0) = 1 0 Relation (2.12) implies that q ∈ K, is a convex domain and Relation (2.11) is equivalent to: For λ = 1, ν = 0, and considering Relation (2.3), we have that (D −1 z I 0 ) ′ (0) = 1 0, and the following corollary is obtained: Corollary 2.3.Let D −1 z I 0 (z) be given by (2.3) and consider q ∈ K as: ) S * , and it is possible that D −1 z I 0 (z) is not univalent in U. □ The next results provide necessary and sufficient conditions for the fractional integral of the Bessel function of the first kind and order ν to be a convex function of positive and negative order, respectively.Theorem 2.4.Let D −λ z I ν (z) as given by (2.1) and consider q ∈ K as: then For λ = 1, ν = 0, and considering Relation (2.3), we get that (D −1 z I 0 ) ′ (0) = 1 0, and the following corollary is obtained: Corollary 2.4.Let D −1 z I 0 (z) be given by (2.3) and consider q ∈ K as: By applying the outcome of Corollary 2.4, we show that Using Relation (2.18) in (2.20), we obtain:  The next results state the necessary and sufficient conditions for the fractional integral of the Bessel function of the first kind and order ν given by (2.1) to be a convex function of positive order α ∈ [0, 1).Theorem 2.6.Let D −1 z I ν (z) as given by (2.1) and consider q ∈ K as: then Proof.First, it is shown that q(z) = 1 1 − z ∈ K, then we get: q ′ (z) = 1 (1 − z) 2 , q ′′ (z) = 2 (1 − z) 3   and Re zq ′′ (z) q ′ (z) (2.25) Since q ′ (0) = 1 0, Relation (2.25) gives that q(z) ∈ K, is a convex domain and differential subordination (2.24) is equivalent to: For λ = 1, ν = 0, and considering Relation (2.3), we get that (D −1 z I 0 ) ′ (0) = 1 0, and the following corollary is obtained: