Geometric characterization of the generalized Lommel–Wright function in the open unit disc

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function


Introduction and mathematical preliminaries
It is widely known that many functions could be called "special".These include certain elementary functions like the exponential, trigonometric, hyperbolic functions and their inverses, logarithmic functions and poly-logarithms, but the class also expands into transcendental functions like Bessel, Lamé, and Mathieu functions.Some of them play a supplemental role, while others, such as the Bessel and Legendre functions, are of primary importance.These functions appear as solutions of the differential equations and systems used as mathematical models of scientific and other phenomena, particularly those sys-tems that change with time and space.We will restrict our present study to the generalized Lommel-Wright function, which could be of particular interest in concrete problems in mechanics, physics, astronomy, and engineering.
Geometric function theory is an area of complex analysis that investigates the geometric properties of analytic functions.It is a mathematical field characterized by a combination of geometry and complex analysis, and its origin began in the nineteenth century.More recently, geometric function theory has become highly significant as the way of algebraic geometry; in addition, the function theory on compact Riemann surfaces has found some results by creating a finite-gap solutions to nonlinear integrable systems, which can be an area of mathematics with a link to mathematical physics.
The theory of univalent functions is one of the greatest and most interesting fields in geometric function theory.Its origin starts from 1851, when the well-known mapping theorem was constructed by Riemann in his doctoral thesis, and which can be regarded as one of the most useful theorems in classical complex analysis.From a planar topology perspective, it is well known that there exist simply connected domains with rough boundaries, and for these domains, there are no clear homeomorphisms between them.However, the Riemann mapping theorem states that such simply connected domains are not only homeomorphic but are also biholomorphic.The Riemann mapping theorem states that if D is a nonempty domain that is a simply connected open subset in the complex plane C, then there exists an injective and holomorphic mapping f that maps D onto the open unit disc U := {z ∈ C : |z| < 1}.This function is known as the Riemann mapping.Nevertheless, his proof was incomplete, while the proof was given completely in 1912 by Carathéodory using the Riemann surfaces.It was simplified by Koebe after two years in a way that did not require these (see, for example, [1,11,13,20,23]).
In more recent years, significant efforts have been made to study the geometric properties of certain (normalized) special functions such as close-to-convexity, starlikeness, and convexity mostly within U. For additional details, we refer, for example, to [18,[24][25][26] for hypergeometric function, to [8,9] for Bessel function, to [21,30,33] for generalized Struve function, to [29] for Lommel function, to [32,34] for generalized Lommel-Wright function, and to [17] for Fox-Wright function.In addition, some radii problems for the Bessel, q-Bessel, Struve, and Lommel functions of the first kind were investigated in [2][3][4][5][6][7] and in the references therein.It was shown that these radii are actually solutions of some transcendental equations.These results could be important to deduce some of the geometric properties of complex functions.
The contents of the present paper are summarized as follows.We outline first various well-known mathematical facts that will be used in the subsequent sections.Moreover, we examine the geometric properties of J ν,m λ,μ , including the starlikeness and convexity of order α, 0 ≤ α < 1, using the mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by [16].In addition, we discuss the starlikeness and convexity of order zero for J ν,m λ,μ , and it turns out that they are useful to extend the range of validity for the parameter λ where the leading concept of the proofs comes from some technical manipulations by [19].Our results improve, complement, and generalize some well-known (nonsharp) estimates.
An analytic function f is called univalent (or schlicht) in a domain D ⊂ C, which is a subset of the complex plane, if it is injective in D. Without loss of generality, we will assume that f is normalized by the conditions f (0) = f (0) -1 = 0 and is defined on U, that is, an analytic function having the Maclaurin series expansion of the form This class of functions is denoted by A, while the subclass of A consisting of univalent functions in U is denoted by S. The most classic example of a function in S is the Koebe function, that is, It maps the unit disc U onto C slit along the negative real axis from -1/4 to -∞, i.e., It is well known that this function plays the extremal role in many problems in the univalent function theory.

Besides, if g
The aforementioned definition of the convolution arises from the integration (see [11]) We are now in a position to recall the most important subclasses of the class of analytic functions, which can be regarded as the cornerstone in the theory of univalent functions, that is, the subclasses of starlike and convex functions (these classes were introduced by Robertson in 1936).
We are now in a position to deduce certain special cases of the generalized Lommel-Wright function.If we set m = 1 in (1.2), we get the Bessel-Maitland function introduced by Pathak [22], which has the form for μ > 0 and λ, ν ∈ C. Putting λ = 1/2 and m = μ = 1 in (1.2), we have the Struve function defined by For λ = 0 and m = μ = 1 in (1.2), we get the Bessel function that has the power series expansion In the following part of the paper, we need the next definition.

Definition 1.1
The normalized form of the combination of generalized Lommel-Wright function is defined by where the function J ν,m λ,μ satisfies the differential equation J ν,m λ,μ (z , and μ ∈ N 0 .Clearly, J ν,m λ,μ can be written as where (r) k denotes the Pochhammer symbol given by (r Remark 1.1 1.First, we will determine sufficient conditions such that J ν,m λ,μ given by (1.4) is well defined.
2. According to the definition of Pochhammer symbol, we should assume that kμ ∈ N for all k ∈ N, that it is equivalent to μ ∈ N.
3. We should assume that the denominator of the above definition formula is not vanishing for any k ∈ N, which is equivalent to 4. Moreover, we should prove that the power series of (1.4) converges in the whole open unit disc U. The radius of convergence of this power series is Using assumptions (1.5) and the fact that μ ∈ N, the second of the above limits is +∞, while the first one is Therefore, R ≥ 1 if and only if m ≥ 0 and (1.5) is satisfied.Concluding, the power series defined by (1.4) is correctly defined and converges in U only if we make the following assumptions: 2 Sufficient conditions for starlikeness and convexity of order α This section aims to investigate a fascinating aspect regarding the geometric properties of the function defined by (1.4), such as starlikeness and convexity of order α, 0 ≤ α < 1, inside the open unit disc using mathematical induction, as well as an estimation for the upper and lower bounds for the gamma function inspired by [16].Our results improve, complement, and generalize some well-known (nonsharp) estimates given in the literature.
or, equivalently, ) By making use of the maximum modulus principle of an analytic function as well as the triangle inequality, together with assumptions (1.6) and the fact that where we assumed in addition, and according to the above assumptions, that λ > 0 and λ + ν > 0. Differentiating logarithmically both sides of (2.4) to get where ψ is the well-known digamma function defined by ψ(z) := (z)/ (z).Assume in addition that λ ∈ N to use in the further proof the induction method.Now, to prove that G(λ) < 0 for all λ ∈ N, we will use the mathematical induction.For λ = 1, we have Using the fact that ν ≥ 0 and since we already know that t ≥ 1, μ ≥ 1, we have ν + tμ + 2 ≥ tμ + 2 ≥ t + 2 ≥ 3, and using the fact that the digamma function ψ is a strictly increasing function on (0, +∞), it follows that and we will prove that (5/3) -(m + μ)ψ(3) < 0. For this purpose, using the relation and the fact that ψ(1) = -γ , where γ is the Euler-Mascheroni constant given by we obtain Further, assuming that G(λ 0 ) < 0 for some λ 0 ∈ N and using (2.6), we have where t ≥ 1, ν ≥ 0, μ ≥ 1, and m ≥ 0. It follows that G(λ 0 + 1) < G(λ 0 ) < 0, therefore G(λ) < 0 for all λ ∈ N. The well-known relation leads to F(t) > 0 for all t ∈ [1, +∞).
Consequently, the right-hand side of (2.3) implies that and so On the other hand, from the maximum modulus theorem of an analytic function, it finds and the above inequality could be rewritten as where F is defined by (2.4), since we already proved that F is a strictly decreasing function on [1, +∞), it follows that G is also a strictly decreasing on the same interval.Therefore, the above inequality leads to From inequalities (2.9) and (2.10) we deduce that and a simple computation shows that the right-hand side of the above inequality is less or equal than 1α if and only if (2.1) holds, or equivalently (2.2).
If we define the function L : [1, +∞) → R by L(t) := H(t)/t, where H is given by (2.14), since we already proved that H is a strictly decreasing function on [1, +∞), it follows that L is also strictly decreasing on the same interval.Therefore, the above inequality implies that (2.17) From inequalities (2.16) and (2.17) we conclude that and a simple computation shows that the right-hand side of the above inequality is less or equal to 1α if and only if (2.11) holds, which is equivalent to (2.12).Using Theorem 1 of [27], we obtain in the following result a sufficient condition for the parameters λ ≥ 1, μ, m ∈ N, and ν ≥ 0 such that J ν,m λ,μ ∈ S * (α), which extends Theorem 2.1, where we assumed that λ ∈ N.
Using similar reasons, the next theorem gives us sufficient conditions such that J ν,m λ,μ ∈ K(α) for a much weaker assumption on λ, that is, only λ ≥ 1.

Theorem 2.4 Suppose that
or, equivalently, Proof As is well known, a function f of the form (1.1) belongs to the class K(α) if and only if zf (z) ∈ S * (α).Since according to [27,Theorem 1], to prove our result, we will show that ∞ k=2 (kα)|B k | ≤ 1α.A simple computation leads to the fact that this inequality is equivalent to Using that λ > -1 and μ ∈ N, it follows
First, we would like to show that U(t + 1) < U(t) for t ≥ 1, hence we will try to find a negative upper bound for the difference Using again the double inequality (2.21), from relation (2.29) it follows that where Since λ ≥ 1, μ ∈ N, m ≥ 1, and ν ≥ 0, we get where, using again the above mentioned assumptions, we deduce that Since A(y) has the same form like in the proof of Theorem 2.3, using the MAPLE™ computer software, we obtain A(y) > 1.2381, y ≥ μ + 2, and from inequality (2.31) we get where and according to (2.32) and (2.30) it follows that U(t + 1) < U(t) for t ≥ 2.
To show that the last inequality holds also for t = 1, we should prove that and ν ≥ 0 or, equivalently, Since λ ≥ 1, μ ∈ N, m ≥ 1, and ν ≥ 0, we have Hence, the inequality U(t + 1) < U(t) holds also for t = 1, therefore For the second term of sum (2.28), we see that for the previously defined function V we have and because we already proved that Consequently, since N ⊂ {1}∪[2, +∞), like in the proof of Theorem 2.3, the above results yield that It follows that for both terms of the sums that appeared in (2.28) we have and and from assumption (2.27) it follows that Using Theorems 2.3 and 2.4, in the next two examples we find the order of starlikeness and convexity for the functions J 3,2 1.5,2.5 and J 2,2 0.2,3.5 , and we emphasize that λ / ∈ N, hence these results cannot be obtained from Theorems 2.1 and 2.2, respectively.Remark 2.1 1.We could see that Theorems 2.3 and 2.4 are more general than Theorems 2.1 and 2.2, respectively.That is because in the first case we replace the assumption λ ∈ N with λ ≥ 1, while in the second case the assumptions λ ∈ N and m ∈ N \ {1} were replaced by λ ≥ 1 and m ∈ N only.

Remark that
under the assumptions of Theorems 2.1 and 2.3, hence if the parameters satisfy the conditions of these two theorems, then J ν,m λ,μ ∈ S(2/3).Also, and min α c = 0 under the assumptions of Theorems 2.2 and 2.4, respectively.Thus, if the parameters satisfy the conditions of Theorem 2.2, then J ν,m λ,μ ∈ K(2/3), while if they satisfy the assumptions of Theorem 2.4, then J ν,m λ,μ ∈ K(0) =: K.

Sufficient conditions for starlikeness and convexity
In this section we give sufficient conditions for the starlikeness and convexity of J ν,m λ,μ by using the next results, respectively.
Remark that for k = 1 the above result was previously obtained by [28,Theorem 3], and we will use it in our next first result.
then |f (z)/f (z)| ≤ 1, z ∈ U, and hence f ∈ K.This result is sharp. and ) Proof To prove the above result, we will use Lemma 3.1 for the particular case k = 1.Thus, to obtain J ν,m λ,μ ∈ S * , it is sufficient to show that Like in the proof of Theorem 2.1, from the assumptions of the parameters we have that On the other hand, by using the left-hand side of the inequality (see [14, Lemma 1]) followed by the left-hand side of (see [12, ineq. (13)]) since λ ≥ 0, m ≥ 0, and μ ∈ N, it follows that under assumption (3.1).Since K is a strictly decreasing function on [1, +∞), then K(t) ≤ K(1) < 0, t ≥ 1, and using that K(t) > 0, t ≥ 1, relation (3.5) leads to K (t) < 0, t ≥ 1.This last inequality implies that K is also a strictly decreasing function on [1, +∞), and from (3.4) we get A simple computation shows that under assumption (3.2) the right-hand side of the above inequality is less or equal than 2/ √ 5. Thus, according to (3.3), the required result follows. and Proof To use Lemma 3.2 for k = 1, we should prove that under our assumption we have Using similar computations like in the proof of Theorem 2.2, we get Like in the proof of Theorem 3.1, using (3.6), (3.7) and the facts that m, μ ≥ 0, we obtain under assumption (3.8).Since Q is a strictly decreasing function on [1, +∞), then Q(t) ≤ Q(1) < 0, x ≥ 1, and using that Q(t) > 0, t ≥ 1, relation (3.12) yields that Q (t) < 0, t ≥ 1.This last inequality implies that Q is also a strictly decreasing function on [1, +∞), and from (3.11) we get Consequently, assumption (3.9) implies that the right-hand side of the above inequality is less or equal than 1/2, and according to (3.10), we obtain our result.

Concluding remarks and outlook
In the present section the highlights of the paper are listed below: 1.In Theorem 2.1 we have used the principle of mathematical induction to generate the starlikeness of order α s , given by (2.1), for the function J ν,m λ,μ (z) defined by (1.4) for μ ∈ N, λ ∈ N, ν ≥ 0, and m ∈ N. Further, in Theorem 2.2, the mathematical induction was also used to obtain the convexity of order α c for μ ∈ N, λ ∈ N, ν ≥ 0, and m ∈ N \ {1}; 2. In Theorems 2.3 and 2.4, an estimate for the upper and lower bounds for the gamma function inspired by [16] has been used to evaluate the orders α s for μ ∈ N, λ ≥ 1, ν ≥ 0, m ∈ N, and α c for μ ∈ N, λ ≥ 1, ν ≥ 0, and m ∈ N; 3. It could be seen that in [31] and [34] the authors investigated the orders of starlikeness and convexity of order α s and α c , respectively, using some well-known estimation for gamma, digamma, and Fox-Wright functions.It is worth mentioning that our results in this paper slightly improve the results in [31] and [34]; 4. Finally, the starlikeness and convexity of order zero for J ν,m λ,μ are studied using some technical manipulations proved by [19] that if f (z) = z + a n+1 z n+1 + • • • , n ≥ 1 is analytic in U and |f (z) -1| < (n + 1)/ (n + 1) 2 + 1, z ∈ U, and |f (z)| ≤ n/(n + 1), z ∈ U, then f ∈ S * and f ∈ K, respectively.These results are useful to extend the range of validity for the parameter λ to λ ≥ 0.
starlike with respect to an interior point z 0 ∈ D if the line segment that joins z 0 to any other point of D lies entirely in D. In particular, if z 0 = 0, then the domain D is called starlike domain.A function f ∈ A belongs to the class S * if and only if Re(zf (z)/f (z)) > 0, z ∈ U.The Koebe function and its rotations are an example of starlike functions, and this function is extremal for the class S * .Moreover, if f (U) is a convex domain, then f ∈ S is called convex, denoted by K.It is well known that the domain D ⊂ C is convex if the line segment joining any two points of D lies entirely in D. Analytically, the convex functions f ∈ A can be represented as

Figure 4
Figure 4 Figures for Example 3.2