On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series

: The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β - uniformly starlike functions Τ 𝒥𝒥 ( 𝛼𝛼 𝑖𝑖 ) . Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions 𝑓𝑓 ∈ Τ 𝒥𝒥 ( 𝛼𝛼 𝑖𝑖 ) . In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass.


Introduction
The term "symmetry" on the open unit disk : = { ∈ ℂ: || < 1} can relate to rotational, reflection, or inversion symmetry, among other kinds of symmetry.The characteristic known as "inversion symmetry" describes how an open unit disk appears when it is inverted with respect to a certain point.When any complex number  in the disk is inverted with respect to the origin, it yields the complex number − whose inversion is also in the disk, indicating that the open unit disk has inversion symmetry with regard to its center (the origin).The open unit disk, in general, contains a rich set of symmetries that are helpful in several geometric and mathematical situations.Our goal was to investigate other geometric characteristics inside this symmetry area.
If a function maps a disk in the complex plane onto a shape that, with relation to a fixed point on the disk, is star-shaped, it is said to be starlike.Stated differently, a function is said to be starlike if, when subjected to appropriate scaling and rotation, its image is contained inside a star-shaped domain.This domain is created by joining the fixed point to every other point in the domain using straight-line segments.While starlike functions are utilized in geometric function theory and mathematical physics to simulate phenomena like electrostatics [1,2] and fluid flow [3,4], univalent functions are frequently used in geometric function theory to explore conformal mappings and the Riemann mapping theorem.
In number theory, (see [10][11][12][13]), the Lambert series is used for certain problems due to its connection to the well-known arithmetic functions such as where  0 () = () is the number of positive divisors of .
where   () is the higher-order sum of divisors function of .We restrict our attention to the series given by (3).In particular, when  = 1, we write  1 () = ().Here, () is the sum of divisors function that appears in one of the elementary equivalent statements to the well-known Riemann hypothesis.
We distinguish at the outset between the Lambert series and the Lambert W function, which appears naturally in the solution of a wide range of problems in science and engineering [14].
In 1984, Guy Robin [15] proved that Moreover, he proved that the Riemann hypothesis is equivalent to where  = 0.7721 ⋯, is the Euler-Mascheroni constant.
This article makes no attempt to prove or refute the Robin's inequality (5) or the Riemann hypothesis.For more details, we refer interested readers to the articles listed in the references [16][17][18][19][20][21].

Preliminaries
Let  denote the class of analytic functions of the form and  be the subclass of  consisting of univalent (or one-to-one) functions on .Let Τ be the subclass of  consisting of functions of the form The importance of the coefficients given by the power series in (6) emerged in the early stages of the theory of univalent functions.
The focus of this research is to introduce a linear operator to define a new subclass of analytic functions of order  such that 0 ≤  < 1.First, it is necessary to recall the two well-known subclasses of starlike and convex functions of order , as given below: Selectively, when  = 0, the above classes are reduced to their standard definition and are simply called the starlike and convex functions.

Definition 1. A function 𝑓𝑓 ∈ 𝒜𝒜 of the form (6) is starlike with respect to symmetrical points if
We denote by  the class of all such functions.Definition 2. A function  ∈  of the form ( 6) is -uniformly starlike of order , if We denote by (, ) the class of all such functions.Definition 3. A function  ∈  of the form ( 6) is -uniformly convex of order , if We denote by (, ), the class of all such functions.
New subclasses of analytic functions have been introduced for various applications, such as fractional calculus and quantum calculus, by involving some special functions, such as the Mittag-Leffler and Faber polynomial functions [31][32][33].The most common concern in such studies is the inclusion conditions.Alternatively, it means that for a given new subclass, ℋ, we seek a set of useful conditions on the sequence {  } that are both necessary and sufficient for () to be a member of ℋ.
By following the same pattern, this study attempts to apply the Lambert series which has not been so yet considered in the theory of univalent functions.Consequently, this may lead to relevant studies if one considers extending the Lambert series, whose the coefficients are the sum of divisors function to other subclasses of analytic functions.Hence, we can investigate various topics such as Hankel determinants, subordination properties, and Fekete-Szegö inequalities.Furthermore, these results can be extended to multivalent functions and meromorphic functions.In addition, by using the two Robin's inequalities, one of which is analogous to the Riemann hypothesis, we can extend the resulting conclusions of some parts of this work and derive further findings.We can also obtain additional forms of the Mittag-Leffler function, including the exponential function, if we take into account certain values of the parameters in the generalized Mittag-Leffler function given by ( 1) and then study various special cases.
Here, we recall the definition of the Hadamard product (convolution): For a given function  ∈  of the form (6) and  ∈  of the form the convolution ( * ) of the two functions  and  is obtained as follows: Subsequently, we utilize the Lambert series ℒ(), whose coefficients are the sum of divisors function ().The mathematical form is In addition, since  , , does not belong to the class , we consider some normalization by introducing For a function  ∈  of the form (7), we define the linear operator (ℒ, )():  ⟶  as follows: The above linear operator leads us to propose a definition in the following manner: Definition 4. A function  ∈  of the form ( 6) is said to be in the class   () if the function  satisfies the following condition: Finally, we consider functions with negative coefficients  ∈ Τ, similarly to the condition (11), and simply write: Τ  () =   ()⋂Τ.Based on Definition 3 and the subclass Τ  (), the analytic characterization of the function  reduces to the following definition.7) is said to be in the class   () if the function  satisfies the condition (11).

Characterization Property
In this section, we discuss the characterization properties of the members that belong to the new family of analytic functions.The characterization properties include a couple of theorems related to the inclusion of functions, consequent corollaries, and a closure theorem.
Theorem 1.A function  ∈  of the form ( 7) is said to be in the class   () if and only if where .
Proof.To prove the assertion in (12), it is sufficient to show that After adding and subtracting 1 from the right-hand side, we obtain The above expression is bounded by (1 − ), thus proving our assertion.
Conversely, let us assume that  ∈ Τ  (), then (12) yields Letting  → 1 along the real axis results in the inequality Finally, the result is sharp with extremal function  given by Let a function  defined by (7) belong to the class   (), then, Next, we obtain lower bounds for the coefficients   using Robin's inequalities in ( 4) and ( 5), the latter of which we simply refer to as the Riemann hypothesis.
Corollary 2. Let a function  defined by (7) belong to the class   ().If Proof.The proof follows from Corollary 1 and inequality (4).□ Corollary 3. Let a function  defined by (7) belong to the class   ().Assuming that the Riemann hypothesis is true, and Proof.The proof follows from Corollary 1 and inequality (5).
and we obtain the following special cases of the previous results:

The coefficients bound in a Corollary 1 become
Similarly, the lower bounds in Corollaries 2 and 3, respectively, will be given by ,  ≥ 3.

Example 3. Under the same conditions of Example 2, assuming the Riemann hypothesis yields
Theorem 2. Let a function  defined by (7) and be in the class   (), then, the function ℎ states that where   = (1 − )  +   , 0 ≤  ≤ 1 also belongs to the class Τ  ().

□
Theorem 1. again entails that ℎ is a member of   ().
Theorem 4. For two functions   () , ( = 1,2) defined by (14), let  1 () ∈   () and  2 () ∈   ().Then,  1 *  2 ∈   (), where Proof.In view of Theorem 1, it suffices to prove that It follows from Theorem 1 and the Cauchy-Schwarz inequality that Thus, it suffices to find  such that By virtue of (12), it suffices to find  such that which concedes the assertion of our theorem.□ Again, by using the inequalities (4) and ( 5), we establish the next two results.For brevity, we use () and Φ() in the forthcoming results, as indicated below.

The Integral Transform of Class 𝚻𝚻 𝓙𝓙 (𝜶𝜶)
To convert class Τ  () into integral form, we define the following integral transform: where () is a real valued, non-negative, and normalized weight function such that The special case of () is () = Proof.By definition, we have, By applying basic mathematical principles, we derive the following expression: We need to prove that Conversely,  ∈ Τ  () if and only if, This shows that +1 + < 1, and, hence, Eq. ( 21) holds.Thus, the proof is evident.□ Next, we derive the radii of starlikeness and convexity of   () where The result is sharp with extremal function () given in the proof of Theorem 1.

Proof. It is sufficiently fair to confirm that
Considering the left-hand side of the above inequality, we write The last expression is less than 1 −  as This completes the proof.□ Utilizing inequalities (4) and ( 5) again, we receive the following Corollary 12. Let  ∈   ().Then,   () is starlike of order 0 ≤  < 1 in || <  1 , where Corollary 13.Let  ∈   () and let us assume that the Riemann hypothesis is true.Then,   () is starlike of order 0 ≤  < 1 in || <  1 , where Finally, for this section, we have: The result is sharp with extremal function () given in the proof of Theorem 1.
(3) If then the second Hankel determinant satisfies the inequality where , ℳ, and  are given by The result is sharp with extremal function () given in the proof of Theorem 1. Further, let or, equivalently where the function  0 is analytic in the unit disk and has a positive real part, by using the Taylor expansion of  , and , we obtain Now, we have By equating the last two Equations ( 25) and ( 26), we get Therefore, where  =  1 > 0 and , ℳ,  are given by (23).Now by applying Lemma 2, we obtain Now, we may assume, without restriction, that  ∈ [0, 2].Since () ∈ , so | 1 | ≤ 2. We set  = ||, where −1 ≤  ≤ 1 and applying triangle inequality on  2 (2) for all || ≤ 1, we obtain where The inequality  2 ≥ 0 is obvious;  1 ≥ 0 such that One can simply show that (,)  > 0 for  > 0, hence, Υ(, ) is an increasing function and, thus, the upper bound for Υ(, ) corresponds to  = 1 and 16(2)( 4)(3) 2 . ( We simplify as  Which, after simple calculations completes the proof of Theorem 10. □

Conclusions
In this article, we introduce a new subclass of uniformly starlike functions by utilizing the Lambert series, with coefficients derived from the arithmetic function σ().Consequently, we explore the characteristics of the proposed subclass.Furthermore, we discuss several relevant topics, including the Hadamard product, integral transform, and radii of starlikeness and convexity.In addition, we extended some findings by incorporating Robin's inequalities and the Riemann hypothesis.Thus, applying the Lambert series to additional subclasses of analytic functions may lead to significant research outcomes.Consequently, we can conduct research on various subjects, including Fekete-Szegö inequalities and subordination characteristics.Furthermore, multivalent functions and meromorphic functions can be included in the scope of these conclusions.
Generally, if we apply the same methodology as this study and take into account the Lambert series, whose coefficients are the higher-order sum of divisors function   (), and if we investigate various special cases of the Mittag-Leffler function, we can also get more intriguing results.