Abstract and Applied Analysis

Abstract and Applied Analysis / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 7391058 | 8 pages | https://doi.org/10.1155/2020/7391058

Initial Bounds for Certain Classes of Bi-Univalent Functions Defined by Horadam Polynomials

Academic Editor: Kunquan Lan
Received09 Jun 2019
Revised02 Sep 2019
Accepted08 Oct 2019
Published30 Jan 2020

Abstract

The main purpose of this article is to make use of the Horadam polynomials and the generating function , in order to introduce three new subclasses of the bi-univalent function class For functions belonging to the defined classes, we then derive coefficient inequalities and the Fekete–Szegö inequalities. Some interesting observations of the results presented here are also discussed. We also provide relevant connections of our results with those considered in earlier investigations.

1. Introduction

Let be the set of real numbers, be the set of complex numbers and

be the set of positive integers. Let denote the class of functions of the form

which are analytic in the open unit disk . Further, by we shall denote the class of all functions in which are univalent in

It is well known that every function has an inverse defined by

and

where

A function is said to be bi-univalent in if both the function and its inverse are univalent in . Let denote the class of bi-univalent functions in given by (2).

In 2010, Srivastava et al. [1] revived the study of bi-univalent functions by their pioneering work on the study of coefficient problems. Various subclasses of the bi-univalent function class were introduced and nonsharp estimates on the first two coefficients and in the Taylor–Maclaurin series expansion (2) were found in the recent investigations (see, for example, [223]) and including the references therein. The afore-cited all these papers on the subject were actually motivated by the work of Srivastava et al. [1]. However, the problem to find the coefficient bounds on for functions is still an open problem.

For analytic functions and in , is said to be subordinate to if there exists an analytic function such that

This subordination will be denoted here by

or, conventionally, by

In particular, when is univalent in ,

The Horadam polynomials or briefly are given by the following recurrence relation (see [22, 23]):

for some real constants , and

The generating function of the Horadam polynomials (see [23]) is given by

Here, and in what follows, the argument is independent of the argument that is,

Note that for particular values of , and , the Horadam polynomial leads to various polynomials, among those, we list a few cases here (see, [22, 23] for more details):(1)For we have the Fibonacci polynomials .(2)For and we obtain the Lucas polynomials .(3)For and we get the Pell polynomials .(4)For and we attain the Pell-Lucas polynomials .(5)For and we have the Chebyshev polynomials of the first kind.(6)For and we obtain the Chebyshev polynomials of the second kind.

Recently, in literature, the coefficient estimates are found for functions in the class of univalent and bi-univalent functions associated with certain polynomials such as the Faber polynomial [8], the Chebyshev polynomials [6], and the Horadam polynomial [15]. Motivated in these lines, estimates on initial coefficients of the Taylor–Maclaurin series expansion (2) and Fekete–Szegö inequalities for certain classes of bi-univalent functions defined by means of Horadam polynomials are obtained. The classes introduced in this paper are motivated by the corresponding classes investigated in [2, 10, 14, 15].

2. Coefficient Estimates and Fekete–Szegö Inequalities

A function of the form (2) belongs to the class for and if the following conditions are satisfied:

and for

where the real constant is as in (10).

Note that was introduced and studied by Srivastava et al. [15].

Remark 1. When and , the generating function in (11) reduces to that of the Chebyshev polynomial of the second kind, which is given explicitly byin terms of the hypergeometric function

In view of Remark 1, the bi-univalent function class reduces to and this class was studied earlier in [3, 12]. For functions in the class , the following coefficient estimates and Fekete–Szegö inequality are obtained.

Theorem 1. Let be in the class . Thenand for

Proof. Let be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions and such thatand we can writeandEquivalently,andFrom (20) and (21) and in view of (11), we obtainandIfthen it is well known thatThus upon comparing the corresponding coefficients in (22) and (23), we haveandFrom (26) and (28), we can easily see thatandIf we add (27) to (29), we getBy substituting (31) in (32), we obtainand by taking and in (33), it further yieldsBy subtracting (29) from (27) and in view of (30), we obtainThen in view of (31), (35) becomesApplying (10), we deduce thatFrom (35), for we writeBy substituting (33) in (38), we havewhereHence, we conclude thatand in view of (10), it evidently completes the proof of Theorem 1.

For Theorem 1 readily yields the following coefficient estimates for .

Corollary 1. Let be in the class . Thenand for In view of Remark 1, Theorem 1 can be shown to yield the following result.

Corollary 2. Let be in the class . Thenand for

Remark 2. Results obtained in Corollary 1 coincide with results obtained in [15]. For . Corollary 2 reduces to the results discussed in [3, 12].
Next, a function of the form (2) belongs to the class for and if the following conditions are satisfied:and for where the real constant is as in (10).
Note that the class reduces to the classes and as and . In view of Remark 1, the bi-univalent function classes would become the class introduced and studied by Altınkaya and Yalçin [4]. For functions in the class , the following coefficient estimates and Fekete–Szegö inequality are obtained.

Theorem 2. Let be in the class . Thenand for

Proof. Let be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions and such thatandandEquivalently,andFrom (53), (54) and in view of (11), we obtainandIfthen it is well known thatThus upon comparing the corresponding coefficients in (55) and (56), we haveandFrom (59) and (61), we can easily see thatandIf we add (60) to (62), we getBy substituting (64) in (65), we obtainand by taking and in (66), it further yieldsBy subtracting (62) from (60) and in view of (63), we obtainThen in view of (64), (68) becomesApplying (10), we deduce thatFrom (68), for we writeBy substituting (66) in (71), we havewhereHence, we conclude thatwhich in view of (10), evidently completes the proof of Theorem 2.

For Theorem 2 readily yields the following coefficient estimates for

Corollary 3. Let be in the class . Thenand for In view of Remark 1, Theorem 2 yields the following result.

Corollary 4. Let be in the class . Thenand for In view of Remark 1, Corollary 3 yields the following result.

Corollary 5. Let be in the class . Thenand for

Remark 3. The results obtained in Corollary 4 and 5 coincide with results of Altınkaya and Yalçin [4].
Next, a function of the form (2) belongs to the class for and if the following conditions are satisfied:and for where the real constants is as in (10).

This class also reduces to and . In view of Remark 1, the bi-univalent function class would become the class . For functions in the class , the following coefficient estimates are obtained.

Theorem 3. Let be in the class . Thenand for

Proof. Let be given by the Taylor–Maclaurin expansion (2). Then, there are analytic functions and such thatandandEquivalently,andFrom (88) and (89) and in view of (11), we obtainandIfthen it is well known thatThus upon comparing the corresponding coefficients in (90) and (91), we haveandFrom (94) and (96), we can easily see thatandIf we add (95) to (97), we getBy substituting (99) in (100), we obtainand by taking and in (101), it further yieldsBy subtracting (97) from (95) and in view of (98), we obtainThen in view of (99), (103) becomesApplying (10), we deduce thatFrom (103), for we writeBy substituting (101) in (106), we havewhereHence, we conclude thatwhich in view of (10) evidently completes the proof of Theorem 2.

In view of Remark 1, Theorem 3 yields.

Corollary 6. Let be in the class . Then