Multivalent Prestarlike Functions with Respect to Symmetric Points

: A class of p -valent functions of complex order is deﬁned with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefﬁcient estimates and the solution to the Fekete–Szeg˝o problem are obtained for the deﬁned function class. Further, we extended the results using quantum calculus. Several consequences of our main results are pointed out.

In the present section, we define a new differential operator motivated by the concept of convex combination of analytic functions and we use the operator to define presumably a new class of multivalent functions of complex order with respect to symmetric points.
We focus on the coefficient estimates, inclusion results and solution to the Fekete-Szegő problem of the defined function class. In the subsequent section, we have extended the study using quantum calculus.

Remark 1.
In the literature, for p = 1, numerous study of Janowski starlike and convex functions of complex order with respect to symmetric points can be found. Here, we give some recent studies as special cases of Λ m p (b; Ψ; α; λ, δ; X, Y). 1.
For studies pertaining to the classes of functions with respect to symmetric points, refer to [12] and references provided therein.

Inclusion Relationship and Initial Coefficient Estimates
Throughout this section, we let We use the following results to obtain the solution of the Fekete-Szegő problem for the functions that belong to those classes we define in the first section.
p k ζ k ∈ R and v is complex number, then and the result is sharp for the functions Replacing ζ by −ζ in (13), Subtracting (13) and (14), we have the following after-simplification: Integrating the equality (15), we obtain or, equivalently, On summarizing the above discussion, we have the following.

Theorem 1.
Let χ ∈ Λ m p (b; Ψ; α; λ, δ; X, Y) and ℵ(ζ) be defined as in (12), then we have where the odd function L(ζ) is defined by the equality L in Theorem 1, we can obtain the integral representation for the odd function L(ζ) in the class Q b s (X, Y, m). and Additionally, for all µ ∈ C, we have where Q 1 is given by The inequality is sharp for each µ ∈ C.

Subclasses of Analytic Functions Using Quantum Derivative
In this section, we define a q-analogue of the operator T m λ,δ χ(ζ) defined in Section 1. The study of Geometric Function Theory in dual with quantum calculus was initiated by Srivastava [14]. For recent developments and applications of quantum calculus in Geometric Function Theory, refer to the recent survey-cum-expository article by Srivastava [15] and references provided therein. Now, we give a very brief introduction of the q-calculus. We let For χ ∈ Π p , the Jackson's q-derivative operator or q-difference operator for a function χ ∈ Π p is defined by From (27), if χ ∈ Π p , we can easily see that for ζ = 0 and note that lim q→1− D q χ(ζ) = χ (ζ). The q-Jackson integral is defined by (see [16]) provided the q-series converges. Further, we observe that D q I q χ(ζ) = χ(ζ) and where the second equality holds if χ is continuous at ζ = 0. The class of q-starlike functions introduced by Ismail et al. in [17] is defined as the class of functions which satisfies the condition Here, we let S * q denote the class of q-starlike functions. Equivalently, a function χ ∈ S * q , if and only if the subordination condition (see ( [18], Definition 7)) zD q χ(ζ) holds.
The q-analogue of the function K p (δ) defined as in (7) is given by , 0 ≤ δ < 1, p = 1, 2, 3, . . .. Srivastava et al. [18,19] introduced function classes of q-starlike functions related with conic regions and also studied the impact of Janowski functions on those conic regions. Inspired by the aforementioned works on q-calculus, we now define the q-analogue of the operator QT m λ,δ χ(ζ) as follows: The functionp ν,σ (ζ) plays the role of those extremal functions related to the conic domain and is given bŷ is Legendre's complete elliptic integral of the first kind and R (t) is complementary integral of R(t). Clearly,p ν,σ (ζ) is in R, with the expansion of the form we obtain , if ν > 1. (33) Instead of defining the same class of functions defined in Definition 1 involving quantum derivative, we define a class (motivated by the study of [12] (Definition 1.2)) involving additional parameters.

Definition 2.
For u, v ∈ C, with u = v, |v| ≤ 1, let the class Q m p (u, v; b; Ψ; α; λ, δ; X, Y) consist of a function in Π p satisfying the subordination condition For other special cases of our classes, see [12] (p. 264).
Coefficient Estimates of Q m p (u, v; b; Ψ; α; λ, δ; X, Y) We need the following results to establish our main results.  [21] (Lemma 6)). Let the function ℵ q (ζ), defined as in the right hand side of (34), be convex in Ω where the function Ψ is defined as in (3). If r(ζ) = [p] q + ∞ ∑ k=1 r k ζ k is analytic in Ω and satisfies the subordination condition

Lemma 4 (see
Proof. If the function Ψ has the power series expansion (3), then, from (1), we have Since the subordination relation is invariant under translation, the assumption (35) is equivalent to Further, because the convexity of ℵ implies the convexity of ℵ(ζ) − [p] q , from Lemma 3, the conclusion follows (36).

Proof. Let us consider
where h(ζ) = [p] q + ∞ ∑ k=1 r k ζ k is analytic in Ω and satisfies the subordination condition h(ζ) ≺ ℵ q (ζ). Equivalently, (38) can be rewritten as On equating the coefficient of ζ p+n , we obtain On computation, we have Using (36) in the above inequality, we have where a p = 1, Υ 0 [p, m, δ] = 1. Taking n = 1 in (39), we obtain The hypothesis is true for n = 1. Now, let n = 2 in (39); we obtain If we let k = 2 in (37), we have Hence, the hypothesis of the theorem is true for k = 2. Following the steps as in [22] (Theorem 2), we can obtain the desired result using mathematical induction.

Conclusions
Using the Hadamard product, we define a new family of multivalent differential operator involving the convex combinations of analytic functions. Using the newly defined operator, the family Λ m p (b; Ψ; α; λ, δ; X, Y) of multivalent functions of complex order with respect to symmetric points is defined to unify the study of various classes of p-valent functions. Inclusion relationship and solution to the Fekete-Szegő problem for the defined function class are here established.
Further, a more comprehensive class of multivalent functions involving quantum calculus is introduced. Srivastava, in [15] (Equation (9.4)), showed that all the results investigated using quantum derivative (q-derivative) can be translated into the corresponding so called post-quantum analogues ((r, q)-derivative) using a straightforward parametric and argument variation of the following types: D r, q χ(ζ) = D q r χ(rζ) and D q χ(ζ) = D r, rq χ ζ r , (0 < q < r ≤ 1).
Hence, the additional parameter r is unnecessary; therefore, here, we restrict our study with a q-derivative rather than extending it to a (r, q)-derivative. Numerous q-results obtained by various authors are shown as special cases of our main results.