Coefﬁcient Estimates for a Subclass of Analytic Functions Associated with a Certain Leaf-Like Domain

: First, by making use of the concept of basic (or q -) calculus, as well as the principle of subordination between analytic functions, generalization R q ( h ) of the class R ( h ) of analytic functions, which are associated with the leaf-like domain in the open unit disk U , is given. Then, the coefﬁcient estimates, the Fekete–Szegö problem, and the second-order Hankel determinant H 2 ( 1 ) for functions belonging to this class R q ( h ) are investigated. Furthermore, similar results are examined and presented for the functions z f ( z ) and f − 1 ( z ) . For the validity of our results, relevant connections with those in earlier works are also pointed out.

where C is the set of complex numbers, is denoted by H (U). Let A be the subclass consisting of functions f ∈ H (U). We represent the functions class with series representation: Definition 2. (See [2].) A function f ∈ A is said to belong to the class R(h), if it satisfies the following condition: f (z) ≺ z + 3 1 + z 2 .

Definition 4. (See
We note from Definition 4 that the q-difference D q f (z) converges to the ordinary derivative f (z) as follows: for a differentiable function f in a given subset of C. Moreover, it is readily deduced from Equations (1) and (7) that: [n] q a n z n−1 .
Up to date, the study of q-calculus has intensely fascinated researchers. This great concentration is due to its advantages in several fields of mathematics and physics. The significance of the operator D q is quite obvious by its applications in the study of the several subclasses of analytic functions. For example, initially, in 1990, Ismail et al. [5] gave the idea of the q-extension of the class of starlike functions in U. Historically speaking, a foothold usage of the q-calculus in the context of geometric functions theory was effectively invoked by Srivastava (see, for details, [6], p. 347 et seq.). Subsequently, remarkable research work has been done by many authors, which has played an important role in the development of geometric function theory. In particular, Srivastava et al. [7] studied the class of q-starlike functions in the conic region, while the upper bound of the third Hankel determinant for the class of q-starlike functions was investigated in [8]. Moreover, several authors (see, for example, [9][10][11][12]) published a set of articles in which they concentrated on the classes of q-starlike functions related to the Janowski or other functions from several different aspects. Additionally, a recently-published survey-cum-expository review article by Srivastava [13] is potentially useful for researchers and scholars working on these topics. In this survey-cum-expository review article [13], the mathematical explanation and applications of the fractional q-calculus and the fractional q-derivative operators in geometric function theory were systematically investigated. For some more recent investigations about the recent usages of the q-calculus in geometric function theory, we may refer the interested readers to [14][15][16][17][18][19][20][21][22][23][24][25][26][27].

Definition 5.
(See [5].) A function f ∈ A is said to belong to the class S * q if it satisfies the following conditions: and: Then, on account of the last inequality, it is obvious that, in the limiting case q → 1−: the above closed disk is merely the right-half plane and the class S * q of q-starlike functions turns into the prominent class S * . Analogously, on behalf of principle of subordination, one may express the relations in (9) and (10) as follows (see [28]): Now, in order to define the new class R q (h) of analytic functions that are associated with a certain leaf-like domain, we make use of the above-mentioned q-calculus and the principle of subordination between analytic functions and define the following.

Definition 6.
A function f ∈ S is said to be in the functions class R q (h) if it satisfies the condition given by: where: Remark 1. It is easy to see that: lim where R(h) is a function class introduced and studied by Priya and Sharma [2].
The determinant H j (n) has also been considered by several authors in the literature on the subject (see, for example, [8,30,31]). In particular, Noor [32] determined the rate of growth of H j (n) as n → 0 for functions f given by Equation (1) with bounded boundary. Ehrenborg [33] studied the Hankel determinant of exponential polynomials. The Hankel transform of an integer sequence and some of its properties were discussed by Layman [34].

Remark 2.
By giving some particular values to j and n, the Hankel determinant H j (n) is reduced to the following form: We note that H 2 (1) is the well-known Fekete-Szegö functional (see, for instance, [35]). On the other hand, we have: where H 2 (2) is known as the second Hankel determinant.
Until now, very few researchers have studied the above determinants for the function class that is associated with a leaf-like domain. Therefore, in this paper, we are motivated to find estimates of the first few Taylor-Maclaurin coefficients of the functions f of the form (1) belonging to the class R q (h), which is associated with a leaf-like domain. We also consider the estimates of the familiar functionals such as |a 3 − λa 2 2 | and |a 2 a 4 − a 2 3 |. Finally, this work will be generalized and extended to hold true for the functions z f (z) and f −1 (z).

Preliminary Results
Each of the following lemmas will be needed in our present investigation. then: for some x (|x| 1) and: for some z (|z| 1).

Lemma 2.
(See [39].) If p(z) ∈ P, then, for any complex number µ, This result is sharp for the functions p(z) given by: Lemma 3. (See [40].) Let the function p ∈ P be given by (4). Then: This inequality is sharp.

A Set of the Main Results
We begin this section by estimating the upper bound of the Taylor-Maclaurin coefficients for the functions belonging to the class f ∈ R q (h).

Theorem 1.
If the function f ∈ R q (h) has the form (1), then: and: Proof. If we suppose that f ∈ R q (h), then there exists a function w(z) ∈ B such that: together with: We now define a function p(z) by: Then, it is clear that p ∈ P. The last relation can be restated in the following equivalent form: Substitution of w(z) from (18) into (17) yields: From the right-hand side of (16), we find that: [n] q a n z n−1 Equating the coefficients of like powers of z, z 2 , and z 3 from the relations (19) and (20), we get: and: respectively. Thus, by applying Lemma 3 in (21), we obtain (13). Next, Equation (22) can be reduced to the following form: together with: Using (24) in conjunction with Lemma 2, we get (14). Finally, we find from Equation (23) that: Substituting for the values of p 1 and p 2 from (21) and (22) and also by applying Lemma 3, one can obtain the result as in Equation (15). The proof of Theorem 1 is thus completed.

Remark 3.
In the special case, if we let q → 1−, Theorem 1 would coincide with the corresponding result of Priya and Sharma [2].

Theorem 2.
If the function f ∈ R q (h) has the form (1), then: together with: Proof. From (21)-(23), upon substituting for the values of a 2 , a 3 , and a 4 , we have: where Λ(q) is given by (26). Substituting for p 2 and p 3 from Lemma 1, we obtain: We assume that: |x| = t ∈ [0, 1] and Then, using the triangle inequality, we deduce that: We now define: Differentiating F q (p, t) partially with respect to t, we have: which, after some elementary calculation, shows that: implying that F q (p, t) is an increasing function of t on the closed interval [0, 1]. Thus, clearly, the maximum value of the function F q (p, t) is attained at t = 1, which is given by: Finally, we set: Then, since p ∈ [0, 2], it follows that: This completes the proof of Theorem 2.
If we let q → 1−, Theorem 2 yields the following corollary.

The Fekete-Szegö Problem for the Class R q (h)
We first prove the following result.

Theorem 3.
If the function f ∈ R q (h) is of the form (1), then: Proof. From Equations (21) and (22), we have: After some suitable simplification, this last relation can be interpreted as follows: where: Now, taking into account (28) and Lemma 2, we obtain the assertion (27). A closer examination of the proof shows that the equality in (27) is attained for: The proof of Theorem 3 is thus completed

Remark 4.
In the special case, if we let q → 1−, Theorem 3 will yield the corresponding result that was already proven by Priya and Sharma (see [2]).

Estimates of the Second Hankel Determinant
In this section, we prove the following result.
Substituting for p 2 and p 3 and by using Lemma 1, we obtain a 2 a 4 − a 2 3 = q 6 + 4q 5 + 11q 4 + 4q 3 + 11q 2 + 4q + 1 p 1 We now set p 1 = p and assume also, without restriction, that p ∈ [0, 2]. Then, by applying the triangle inequality on (30), with |x| = t ∈ [0, 1], we find that: By assuming further that: Differentiating F q (p, t) partially with respect to t, we have: which implies that, as a function of t, F q (p, t) increases on the closed interval [0, 1]. This means that F q (p, t) has a maximum value at t = 1, which is given by: We now set: Then, since p ∈ [0, 2], it follows that: which completes the proof of Theorem 4.

Remark 5.
If, in Theorem 4, we let q → 1−, we get the corresponding result due to Priya and Sharma [2].

Coefficient Estimates for the Function z f (z)
Let the function G(z) be defined by: We now prove the following result.
Theorem 5. Let the function h(z) be defined by (12). Suppose also that: Then, for any σ ∈ C, it is asserted that: Proof. Since f ∈ R q (h), we have: Equating the coefficients of z and z 2 from (31) and (33), it can be deduced that: and: Thus, on account of (21), (22), (34), and (35), we get: and: Now, for σ ∈ C, we set: where: Thus, by applying Lemma 2 and after some suitable computation, Equation (38) is reduced to (32). The sharpness of the estimate is given by: Our demonstration of Theorem 5 is now complete.
As a special case of Theorem 5, if we let q → 1−, we get the following known result.

Coefficient Estimates for the Function f −1 (z)
Here, in this section, we prove the following result.
The above-asserted estimate is sharp.
Proof. It is well known that every function f ∈ S has an inverse f −1 , which is defined by: By means of the above relation and (1), we find that: f −1 (z + ∞ ∑ n=2 a n z n ) = z.
It is also known that: f −1 (w) = w + ∞ ∑ n=2 d n w n .
This completes our proof of Theorem 5.
As a special case of Theorem 5, if we let q → 1−, we are led to the following known result.