Abstract
In this work, by considering the Chebyshev polynomial of the first and second kind, a new subclass of univalent functions is defined. We obtain the coefficient estimate, extreme points, and convolution preserving property. Also, we discuss the radii of starlikeness, convexity, and close-to-convexity.
1. Introduction
Let be the open unit disk and be the class of analytic functions in , satisfying the normalized conditions:
Thus, each has the following Taylor expansion:
Furthermore, by , we shall denote the family of all functions in that are univalent in . Denote by the subclass of consisting of functions with negative coefficients of the type: see [1].
Many researchers deal with orthogonal polynomials of Chebyshev, see [2, 3] and [4]. The Chebyshev polynomials of first kind and the second kind are defined by
respectively, where , , and is the degree of polynomial.
The polynomial in (1) is connected by the following relations:
We note that if , , then
Also, we have where are the Chebyshev polynomials of the second kind, see [5, 6] and [7].
The generating function of the first kind of Chebyshev polynomial , is given by
For more details, see [8, 9] and [10].
For two functions and , analytic in , we say that is subordinate to in , written if there exists a Schwarz function which is analytic in with
such that , (), see [11].
Also, if is univalent in , then
Furthermore, if and , then the Hadamard product (or covolution) of and is defined by
Now, we consider the following functions which are connected with the Chebyshev polynomial of the first and second kind: where and “” denotes the Hadamard product.
With a simple calculation we conclude that belongs to and it is of the form:
where and .
Definition. For , , , and , we say that of the form (18) is a member of if the following subordination relation holds where , .
Equation (19) is equivalent to the following inequality:
2. Main Results
In this section, we introduce a sharp coefficient bound for the class . Also, the convolution preserving property is investigated.
Theorem 1. The function of form (18) belongs to if and only if
Proof. Let the inequality (21) holds and . We have to prove that (19) or equivalently (20) holds true. But we have
By putting and
the above expression reduces to
Since , by using inequality (21), we get , so .
To prove the converse, let , thus
for all . By for all , we have
By letting , through positive values and choose the values of such that is real, we have
and this completes the proof.
Remark. We note that the function: shows that the inequality (21) is sharp.
Theorem 2. Let be in the class , then belongs to , where
Proof. It is sufficient to show that
By using the Cauchy-Schwarz inequality, from (21), we obtain
Here, we find the largest such that
or equivalently for ,
where is given by (32).
This inequality holds if
or equivalently
where is given by (32), so the proof is complete.
3. Geometric Properties of
In this section, we show that the class is a convex set. Also, the radii of starlikeness, convexity, and close-to-convexity are obtained.
Theorem 3. The class is a convex set.
Proof. It is enough to prove that if for , be in , then the function is also in , where . But, we have Since by Theorem 1, so, . Hence, the proof is complete.
Theorem 4. Let , then (i) is a starlike of order () in where(ii) is convex of order () in where(iii) is close-to-convex of order () in , where
Proof. (i)For , we need to show thatIn other words, it is sufficient to show that By (21), it is easy to see that above inequality holds if This completes the proof of (i). (ii)Since is convex if and only if is starlike, we get the required result (ii)(iii)We must show that . ButThus, if . But by Theorem 1, the above inequality holds true, if Hence, the proof is complete.
4. Conclusions
Univalent functions have always been the main interests of many researchers in geometric function theory. Many studies recently related to Chebyshev polynomials revolved around classes of analytic normalized univalent functions. In this particular work, the geometric properties are obtained for functions in more general class using the Chebyshev polynomials associated with a convolution structure. In this paper, when the parameters being complex numbers could be subject to further investigation. Also, by changing the operator and extending, it may be for future studies.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to and approved the final manuscript.