On classes of ζ -uniformly q -analogue of analytic functions with some subordination results

This article presents new classes of ζ -uniformly q -starlike and q - convex analytic functions of order υ using the principle of q -calculus. The renowned classes of starlike and convex functions are utilized to establish these classes. The investigation of these uniformly classes leads to the study of some geometric notions, including coefficient estimates, sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, some subordination properties involvingintegralmeansinequalitiesandsubordinatesequencesareexamined.


Introduction
We represent the set of analytic functions f in U as A , where and is the open unit disc.The set S ⊂ A comprises of all functions that are both normalized and univalent.We also consider the subset T of the set A of the following analytic function: If the functions f and h belong to the class of holomorphic functions (i.e. they are both analytic), then f is subordinate to h (f ≺ h), if a Schwarz function ϑ exists and satisfying certain conditions ϑ(0) = 0 and |ϑ(z)| < 1, such that f(z) = h(ϑ(z)).
Equivalently, if the inequality below holds: then the functions f and h will be subordinated.
The convolution of any analytic functions f, ∈ A , expressed by f * , is given as where f(z) have been aforementioned in (1) and The renowned starlike function class, denoted by S * , is formulated by Moreover, the convex function class, denoted by C, is formulated by Noting that from inequalities (3) and (4) (see [1]), we have For ρ ∈ C\{0}, Nasr and Aouf [2] established the subsequent class of starlike functions S * (ρ) of order ρ while Wiatrowski [3] defined and verified the class of convex functions C(ρ) of order ρ as follows: The ζ -uniformly of starlike functions US * (υ, ζ ) of order υ is given by and the ζ -uniformly of convex functions UC(υ, ζ ) of order υ is given by where the classes US * (υ, ζ ) and UC(υ, ζ ) with their generalizations provided by Shams et al. [4] (also see [5][6][7][8][9][10], for related findings).

Quantum calculus
Quantum calculus is a fascinating mathematical framework that extends traditional calculus by incorporating principles from quantum mechanics.It introduces a new perspective on differentiation and integration, allowing for the analysis of functions and operators in a quantum-like context.The principle of q-calculus has profoundly influenced the investigation of Geometric Function Theory (GFT) and its significant applications in various areas, such as mathematical science and quantum physics.This principle presents a resemble framework to ordinary calculus but eliminates the need for limits.The notion of q-calculus, involving the derivative and integral, was primarily provided by Jackson [11].The investigation of q-calculus has led to the exploration of other pertinent aspects, such as the examination of special functions (for a more comprehensive review, see [12][13][14][15][16][17][18][19][20][21]).Ismail et al. [22] proposed the notion of q-calculus in the context of GFT.This work has led to the identification of various classes of analytic functions in the disc U, known as the Ma and Minda classes, which exhibit a strong connection to the concept of subordination.Furthermore, the use of q-calculus, specifically fractional q-integral operators, has been employed to build a multitude of analytic functions.Moreover, a considerable body of literature has been dedicated to the study of certain categories of analytic functions in U through the utilization of q-calculus (e.g.see [23][24][25][26][27][28][29][30][31][32]).
This investigation begins with the essential notions and consequently, the analysis for our proposed study of the q-calculus.This article makes the assumption that 0 < q < 1.The next definitions are necessary for a univalent function f:

Definition 2.2 ([11]):
The operator denoted as D q represents the q-derivative and given by Then the q-derivative of f in (1) is supplied by In addition, the notion of q-generalized Pochhammer is supplied by and the definition of the q-Gamma function is Given that the q-derivative is a generalized version of the ordinary derivative.Thus, considering the classes ( 5)-( 8), we introduce novel classes of ζ -uniformly analytic functions that exhibit q-starlikeness and q-convexity in U. We extensively investigate various aspects including coefficient estimates (|a i |), growth and distortion theorems, Hadamard products property, convex linear combinations, and several subordination properties encompassing integral means inequalities and subordinate factor sequences as well.

Definition 3.1: The function
where We also define Example 3.3: The classes S * q (ρ, υ, ζ ) and C q (ρ, υ, ζ ) reduce to various subclasses as follows: , which have been established by Mahmood et al. [38], which are formulated by ( , which have been defined by Seoudy and Aouf [39], which are formulated by

A set of main results
The following theorems determine coefficient estimates |a i | for the classes S * q (ρ, υ, ζ ) and C q (ρ, υ, ζ ) as outlined below: Proof: Let f(z) ∈ T S * q (ρ, υ, ζ ), then from (9) we have Using theorem of maximum modulus, we conclude that Conversely, let the inequality (11) be correct for z ∈ U, then If we let z → 1 − along the real axis, select the values of z for which D q f(z) is real, and then eliminate the denominator from the above inequality, we can observe that for i ≥ 2, −1 ≤ υ < 1, ζ ≥ 0, and ρ ∈ C\{0} with the sharpness of f(z) given by and we find that the sharpness of f as below we can obtain the desired result utilizing an approach resembling that of Theorem 4.1.
Theorems 4.1 and 4.3 lead to the subsequent corollaries, when q → 1 − .
From the fact |z| = r < 1, we get Similarly, we obtain Here, the proof completes.

Theorem 4.7 (Distortion Theorem
Proof: Our argument of this theorem is similar to that of Theorem 4.6.
From the fact |z| = r < 1, we get Similarly, we obtain Hence, the desired outcome can be inferred.
Similarly, we can establish the subsequent growth bounds: Theorem 4.10: Put f 1 (z) = z and where J i ≥ 0 and ∞ i=1 J i = 1.
Proof: Assume that f(z) ∈ T S * q (ρ, υ, ζ ), then Corollary 4.2 gives Conversely, let f(z) be defined as Then Consequently, it may be inferred that Theorem 4.11: Put f 1 (z) = z and where J i ≥ 0 and ∞ i=1 J i = 1.
Proof: By employing the identical technique as demonstrated in Theorem 4.10, we can attain the necessary finding of Theorem 4.11.

Integral means inequalities
To investigate our main findings, we must mention the following Littlewood's Lemma: Equality holds if and only if r = 0 or if r = 0 but constant, or (z) = e iυ z.
Theorem 4.14: Suppose that f(z) ∈ T S * q (ρ, υ, ζ ) and For z = re iθ , we deduce that The proof of the desired result readily follows from the application of Theorem 4.13.
With the same approach used in the proofs of Theorems 4.13 and 4.14, we find Theorem 4.15.

Subordinate sequence properties
This section explores interesting subordination results within the classes of ζ -uniformly of q-starlike S * q (ρ, υ, ζ ) and q-convex C q (ρ, υ, ζ ) of analytic functions.

Lemma 4.19 ([41]):
A complex sequence {η i } ∞ i=1 is said to be subordinating factor sequence iff The constant is the best estimate.
Thus, by Definition 4.18, Theorem 4.20 is correct if the following sequence is a subordinating factor sequence, with a 1 = 1. .
By Lemma 4.19, equivalently Now, for |z| = r < 1, then This establishes the validity of inequality ( 21) and confirms the subordinating result (19) as proven in Theorem 4.20.The inequality (20) follows from ( 19) by taking Now, we express the function P(z) as below Then from (19), we have This demonstrates the best possible of the following constant: .
Theorem 4.21: Suppose f(z) ∈ T C q (ρ, υ, ζ ) and (z) be any other function in the class of convex functions, then The constant is the best estimate.
Proof: The proof is similar to that of Theorem 4.20.

Conclusion
Having as inspiration the results obtained by applying q-calculus on certain analytic functions, we have established in this investigation new classes of ζ -uniformly q-starlike and q-convex functions of order υ.Additionally, we have derived a range of geometric properties, including estimates for coefficients, the sharpness, distortion and growth theorems, and convex linear combinations.Furthermore, we have leveraged the classes of uniformly q-starlike and q-convex functions to investigate various subordination findings, such as integral means inequalities and subordinate sequences.This study serves as a valuable reference for future works and sheds light on new concepts within geometric function theory.Moreover, numerous other geometrical consequences such as the convolution, radius theorems, closer property, partial sums and neighbourhood, which have not yet been addressed and are still open to other researchers.

Theorem 4 . 20 :
Suppose f(z) ∈ T S * q (ρ, υ, ζ ) and (z) are any other function in the class of convex functions, then