Abstract
In this paper, we investigate an interesting class of analytic and biunivalent functions in the open unit disk which is defined using the -derivative operator. We apply the subordination method to the functional of coefficients problem. Furthermore, we obtain the bounds of the certain functional of coefficients for functions in the class , for real and complex parameters. Also, an estimation for the initial Taylor and Maclaurin coefficients of functions in was given.
1. Introduction
The class of all analytic functions was denoted by . Functions belonging to this class can be displayed in the form of the following power series:
The class of univalent functions in normalized with the conditions was represented by .
Since each function which belongs to the class has an inverse, we can easily calculate
Therefore, we conclude that is analytic [1]. So, can also be displayed in the form of a power series as follows: where
A function is called biunivalent in open unit disk , if and are univalent in open unit disk .
is considered a symbol of the class of biunivalent functions . For more information on the class , readers can refer to [2].
In recent years, the subject of quantum calculus has been considered by many researchers. There are many applications of quantum calculus in various branches of mathematics and physics. -derivatives and -integrals by Jackson [3, 4] were first systematically way introduced and studied; also, he was the first to introduce applications of quantum calculus. For more information about quantum calculus, readers can refer to [5–8].
The question that arises is whether it is possible to study concepts from a quantum calculus point of view in articles such as [9–14] that deal with equations numerically and topics related to numerical analysis.
The theory post quantum calculus or ()-calculus operators are used in various areas of science and also in the geometric function theory.
Let The ()-bracket is defined by
Notice that
The ()-derivative operator of a function is given by
For a function , it can be easily seen that and
For definitions and properties of ()-calculus, one may refer to [15].
The ()-derivative operator is known as the -derivative operator and is denoted by for a function and is defined by
For more details, see [3, 4, 16]. Using equations (1) and (8), for and , we obtained where
It is clear that .
Also, the th order -derivative of is defined by
For this results, we refer to [17].
One of the important subclass of analytic functions defined by -derivative is ; the class of -starlike functions consists of that satisfies
Remark 1. For functions , the Alexander theorem [18] was used by Baricz and Swaminathan [19] for defining the class of -convex functions as follows:
Hence, the class of -convex functions consisting of satisfies
Also, this study introduces a concept of the fractal derivative of a function with respect to a fractal measure : the fractal derivative (15) (see [20]) differs from the standard fractional derivative in that is does not involve the integral convolution and is local in nature.
The elementary physical concepts such as velocity in a fractal spacetime can be defined by where represents time-space fabric having scaling indices and .
Like the fractional derivative, the fractal derivative exists under a fractal metric spacetime [21].
For instance, exists while does not [20].
For more information about fractal and fractional calculus, readers can refer to [22–26].
We say is subordinate to and is written as , if there exists a function belonging to class of Schwartz functions which are satisfying and , such that where and are analytic in
Let be univalent function, , if and only if and .
One of the references that deals with differential subordination is the book of Miller and Mokanu; researchers can refer to [27].
In the subordination method, we can use fixed point theory. In definition of subordination exists Schwarz function , such that Researchers for more information about fixed point theory can refer to [28–30]. It seems quantum calculus has interesting applications in fixed point theory.
The class of functions which are analytic in and satisfying the following conditions and denoted by
In the sequel, for obtaining our main results, we used the following lemma [18].
Lemma 2. If , then for each .
Lemma 3. If , then for some with
Definition 4. A function of the form (1) is said to be in the class if where , , and .
Theorem 5. If a function of the form (1) is in the class , then where and
In this work, we investigate on the classes and of functions which are analytic and biunivalent in the open unit disk . We obtain the bounds of the certain inequality for the class and provide estimates for the initial Taylor and Maclaurin coefficients of functions in .
2. Estimation of the Bounds of Certain Functional of Coefficients for
In this section, we obtain the bounds of certain functional of coefficients for .
Theorem 6. If and , then where
Proof. Our proof begins with the observation of the subordination method for conditions (19) and (20). It follows that
respectively, where
for .
Now, equating the coefficients in (24) and (25), we see that
From (27) and (29), we conclude that
Also, using (28) and (30), we obtain
Subtracting (30) from (28) yields
Thus, it follows from (33) and (34) that where is given in (23). So, applying the triangle inequality and Lemma 2, we complete the proof by considering the following cases.
Case 1.
Case 2.
Remark 7. Note that letting in Definition 4, we obtain the class introduced by Bulut [31]. Also, we obtain the class introduced by Sirivastava et al. [2].
Corollary 8. If a function of the form (1) belongs to the class and , then where
Theorem 9. If and , then where
Proof. From (31), (32), and (34), we deduce that
Now, Lemma 3 shows that
In (43), by (44), we conclude that
By choosing , we may assume without restriction that . Now, by using the triangle inequality and letting and , we obtain
for :
For ,
which proves the existence of a critical point at , that is, at
Since
is a strictly decreasing function of . Now, (47) becomes
Also, since
is an increasing function of . Thus, it follows from (47) that
This completes the proof.
Corollary 10. If a function of the form (1) belongs to the class and , then where
3. Coefficient Bounds for Functions in
Definition 11. A function of the form (1) is said to be in the class if where , , and .
Theorem 12. If a function of the form (1) is in the class , then where
Proof. It follows from (55) and (56) that
where
are in .
Now, equating the coefficients in (59) and (60), we obtain
From (62) and (64), we deduce that
Also, using (63), (65), and (67), we obtain
Therefore,
Applying Lemma 2 to the equality above, we obtain the desired estimate for , as asserted in (57).
Next, in order to find the bound for , we subtract (65) from (63). This gives
It follows from (66), (67), and (70) that
Applying Lemma 2 to the equality above, we obtain the desired estimate for , as asserted in (58).
Remark 13. Note that letting in Definition 11, we obtain the class introduced by Bulut [31]. Also, we obtain the class introduced by Srivastava et al. [2].
In the next section, we introduce some methods for solving linear as well as nonlinear ordinary/partial differential equations. For example, the Exp-function method, homotopy perturbation method (HPM), and variational iteration method (VIM) are introduced.
4. Some Methods for Solving Linear and Nonlinear Differential Equations
4.1. Exp-Function Method
Suppose we are given a partial differential equation (PDE) for a function in the form where is a polynomial in its arguments.
The main idea of the Exp-function method for solving equation (73) proceeds in the following steps.
Step 1. Look for traveling wave solutions of equation (73) by taking the wave transformation: into account, where and are real nonzero constants.
Step 2. Substitution of equation (74) into (73) yields an ordinary differential equation (ODE) for If possible, integrate the resulting ODE term by term one or more times. This introduces one or more integration constants.
Step 3. Introduce the form where , and are positive integers (to be determined). The coefficients and are arbitrary real constants (to be specified).
Step 4. Determine the highest order nonlinear term and the linear term of highest order in the ODE from Step 2, and express them in terms of (75). Then, in the resulting terms, balance the highest order Exp-function to determine and and the lowest order Exp-function to determine and .
Step 5. With , and as determined in Step 4, substitution of (75) into the ODE from Step 2 yields an algebraic equation involving the integer powers of . Equating the coefficients of each power of to zero results in a (highly nonlinear) system of algebraic equations for , and . If the original PDE contains some arbitrary parameters, these will, of course, also appear in the system.
Step 6. The original PDE has a solution in the form (75) if and only if there is a real nontrivial solution to the resulting system of algebraic equations, and using a computer can be very helpful.
For more information about the Exp-function method, readers can refer to [32–38].
4.2. Homotopy Perturbation Method
The homotopy perturbation method is a semianalytical technique for solving linear and nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations [39–42].
To illustrate the basic idea of the HPM, we consider the following differential equation: subject to the boundary condition where represents a general differential operator, is a boundary operator, is the boundary of the domain , and is a known analytic function.
The operator can be decomposed into two parts viz linear and nonlinear . Therefore, equation (76) may be written in the following form:
Also, we have where is the embedding parameter.
Using homotopy technique, proposed by He [41], we construct a homotopy to equation (79) which satisfies
Here, is an initial approximation of equation (80) satisfying the given conditions.
By substituting and in equation (80), we may get following equations, respectively,
As changes from zero to unity, changes from to . In topology, this is called deformation, and and are homotopic to each other. Due to the fact that is a small parameter, we consider the solution of equation (80) as a power series in as below:
The approximate solution of equation (76) may then be obtained as
The convergence of the series solution (84) has been given in [41].
For more information about the homotopy perturbation method, readers can refer to [43–45].
4.3. Variational Iteration Method
The variational iteration method is one of the well-known methods for solving linear and nonlinear ordinary as well as partial differential equations [46, 47]. He in [48] developed the VIM and successfully applied to ordinary and partial differential equations.
To illustrate the basic idea of the VIM, we consider following the general nonlinear system as where is a linear operator, is a nonlinear operator, and is the given continuous function. The correction functional for the system (85) where is a Lagrange multiplier, which can be identified optimally via variational theory, is the th approximate solution, and denotes a restricted variation, i.e.,
The initial approximation can be chosen freely if it satisfies the given conditions. The solution is approximated as
For more information about the variational iteration method, readers can refer to [51–53].
5. Conclusion
The coefficient estimation problems have always been among the main interests of researchers in the study of univalent and biunivalent functions. Many studies related to this problem concern normalized analytic functions. In this paper, the bounds of the functional of coefficients and initial coefficient estimates were obtained for functions in , and estimates were found for the initial Taylor and Maclaurin coefficients of functions in . In a future research, we may obtain the bounds of the Teoplitz and Hankel determinants for classes of functions which are defined in terms of the ()-derivative and -derivative operators.
Data Availability
No data were used.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors read and approved the final manuscript.
Acknowledgments
The research on “Subordination method for the estimation of certain subclass of analytic functions defined by the -derivative operator” by Khon Kaen University has received funding support from the National Science, Research and Innovation Fund.