A Certain Subclass of Uniformly Convex Functions with Negative Coefficients Defined by Gegenbauer Polynomials

In this paper, we introduce a new subclass of uniformly convex functions defined by Gegenbauer polynomials with negative coefficients. For functions in the class T S, we attain coefficient bounds, growth distortion properties, extreme points and radii of close-to-convexity, starlikeness and convexity. For this class, we also produced modified Hadamard product, convolution, and integral operators.


Introduction
Let A indicate the class of all functions u(z) of the form in the open unit disc E = {z ∈ C : |z| < 1}. Let S be a subclass of A that contains univalent functions and satisfies the usual normalization condition u(0) = u (0) − 1 = 0. The subset of A comprising of functions u(z) that are all univalent in E is represented by S. A function u ∈ A is a starlike function of the order υ, 0 ≤ υ < 1, if it fulfils We indicate this class with S * (υ) . A function u ∈ A is a convex function of the order υ, 0 ≤ υ < 1, if it fulfils We indicate this class with K(υ). The regular classes of starlike and convex functions in E are S * (0) = S * and K(0) = K, respectively. Let T indicate the class of functions analytic in E that are of the form and let T * (υ) = T ∩ S * (υ), C(υ) = T ∩ K(υ). Silverman [21] has thoroughly studied the class T * (υ) and related classes, which have some interesting properties. In [2,5], and others have recently looked into several subclasses of T.
For u ∈ A given by (1) and g(z) given by their convolution indicate by (u * g), is specified as Note that u * g ∈ A.
(2). If a function u ∈ A satisfies the condition, it is said to be in the class SP(ρ, γ), uniformly γ− starlike function.
Szynal [24] introduced class T (λ ), λ ≥ 0 and examined it as the subclass of A consisting of functions of the form where and µ is a probability measure on the interval [−1, 1]. The collection of such measures on [a, b] is denoted by P [a,b] . In (9), the Taylor series expansion of the function gives and the coefficients for (10) were given below: where c λ η (t) indicates the Gegenbauer polynomial of degree η. Varying the parameter λ in (10), we obtain the class of typically real functions studied by [6,14,20] and [22].
Let G λ ,t :A → A specified in terms of convolution by We can now describe a new subclass of functions belonging to the class A by using the linear operator G λ ,t . Definition 1.1. For −1 ≤ υ < 1 and ρ ≥ 0, we let T S(υ, ρ, λ ,t) be the subclass of A consisting of functions of the form (4) and fulfiling the analytic condition for z ∈ E.
The class T S(υ, ρ, λ ,t) can be reduced to the class studied earlier by Ronning [15,16] by suitably specialising the values of υ and ρ. The primary aim of this paper is to examine some common geometric function theory properties such as coefficient bounds, distortion properties, extreme points, radii of starlikeness and convexity, Hadamard product, and convolution and integral operators for the class.

Coefficient Bounds
We get a required and adequate condition for function u(z) in the class T S(υ, ρ, λ ,t) in this section. To find the coefficient estimates for our class, we use the approach proposed by Aqlan et al. [4].
Proof. We have f ∈ T S(υ, ρ, λ ,t) if and only if the condition (13) satisfied. Upon the fact that Equation (13) may be written as Now, we let Then (15) is equivalent to For E(z) and F(z) as above, we have and similarly which yields (14).
On the other hand, we must have Upon choosing the values of z on the positive real axis where 0 ≤ |z| = r < 1, the above inequality reduces to Letting r → 1 − , we get the desired result. Finally the result is sharp with the extremal function u given by

Growth and Distortion Theorems
Theorem 3.1. Let the function u defined by (4) be in the class T S(υ, ρ, λ ,t). Then for |z| = r Equality holds for the function Proof. Since the other inequality can be explained using identical reasoning, we just prove the right hand side inequality in (17). In view of Theorem 2.1, we have Since, which yields the right hand side inequality of (17). Let the function u defined by (4) be in the class T S(υ, ρ, λ ,t). Then for |z| = r Equality holds for the function given by (18).
Proof. Since f ∈ T S(υ, ρ, λ ,t) by Theorem 2.1, we have that Thus from (19), we obtain which is right hand inequality of Theorem 3.2.
On the other hand, similarly and thus proof is completed. .
Equality holds for the function given by (18).

Radii of Close-to-Convexity, Starlikeness and Convexity
A function u ∈ T S(υ, ρ, λ ,t) is said to be close-to-convex of order δ if it satisfies Also A function u ∈ T S(υ, ρ, λ ,t) is said to be starlike of order δ if it satisfies Further a function u ∈ T S(υ, ρ, λ ,t) is said to be convex of order δ if and only if zu (z) is starlike of order δ that is if Let u ∈ T S(υ, ρ, λ ,t). Then u is close-to-convex of order δ in |z| < R 1 , where The result is sharp with the extremal function u is given by (16).
But Theorem 2.1 confirms that Hence (21) will be true if We obtain Theorem 5.2. Let u ∈ T S(υ, ρ, λ ,t). Then u is starlike of order δ in |z| < R 2 , where The result is sharp with the extremal function u is given by (16).
Hence, by using (22) and (24) will be true if which completes the proof.
By using the same approach in the proof of Theorem 5.2, we can show that zu (z) u (z) − 1 ≤ 1 − δ , for |z| < R 3 , with the aid of Theorem 2.1. Thus we have the assertion of the following Theorem 5.3. Theorem 5.3. Let u ∈ T S(υ, ρ, λ ,t). Then u is convex of order δ in |z| < R 3 , where The result is sharp with the extremal function u is given by (16).

Inclusion Theorem Involving Modified Hadamard Products
For functions in the class A, we define the modified Hadamard product (u 1 * u 2 )(z) of u 1 (z) and u 2 (z) given by We can prove the following. Theorem 6.1. Let the function u j , j = 1, 2, given by (25) be in the class T S(υ, ρ, λ ,t) respectively. Then (u 1 * u 2 )(z) ∈ T S(υ, ρ, λ ,t, ξ ), where Proof. Employing the approach used earlier by Schild and Silverman [19], we need to find the biggest ξ such that Since u j ∈ T S(υ, ρ, λ ,t), j = 1, 2, then we have by the Cauchy-Schwartz inequality, we have Thus it is sufficient to show that Note that λ ,t, η) .
Consequently, we need only to prove that or equivalently is an increasing function of η, η ≥ 2, letting η = 2 in last equation, we obtain Finally, by taking the function given by (18), we can see that the result is sharp.

Conclusion
This research has introduced a new subclass of uniformly convex functions defined by Gegenbauer polynomials with negative coefficients and studied some basic properties of geometric function theory. Accordingly, some results related to coefficient estimates, growth and distortion properties, radii of starlike and convexity and convolution properties have also been considered, inviting future research for this field of study.