Abstract
Beta function has some applications in differential equations and other areas of sciences and engineering where certain definite integrals are used. However, its applications to univalent functions have not been explored based on the available literature. In this work, therefore, the authors defined a univalent function associated with the beta function with . Some geometric properties of the function are discussed.
1. Introduction and Preliminaries
Let denote the class of functions of the form
which are analytic in the open unit disk and normalised by and and let denote the subclass of consisting of functions whose nonzero coefficients, in the second on, are negative which is defined by
Quite a number of life problems resulted from differential equations when formulated mathematically. These equations come in the form of partial differential equations (PDE) and ordinary differential equations (ODE). The ODE can also come in different orders such as first order or second order depending on the number of factors. ODEs could be linear equations or nonlinear equations. There are different methods of analytical solutions of the secondorder differential equations and the peculiarity of the differential equations matter. For linear ODE, for instance, the most effective method of obtaining the solutions once the variable coefficients are analytic functions is the power series solution. They are called power series methods because the solutions come in certain series forms. The series solution is effective for most differential equations. Special functions such as Bessel equation, Hermite equation, Frobenius equation, hypergeometric equation, Laguerre equation, and Legendre equation are in the form of differential equations. Other special functions such as Chebyshev polynomials, Hermite polynomial, Legendre polynomial, factorial function, gamma function, and beta function are also useful techniques of solving problems in mathematics. Both beta and gamma functions are called the Euler’s integrals. A beta function can be in form of polynomial, polar, or the gamma function. It is often referred to as the Eulerian integral of the first kind while the gamma function is the Eulerian integral of the second kind.
The beta function denoted by is defined by which converges for and . There are three cases of beta function which include: (1)When is a positive integer(2)When is a positive integer(3)When both and are positive integers
Using case 3, we can express a beta function in the form of the gamma function as follows: which indicates the symmetric property of the beta function [5, 6].
By ratio test, let
Then, converges if
That is, and diverges for
That is, or which implies that converges only for case 3 and diverges for either case 1 or case 2.
Suppose , then the radius of convergence or . There are three possible cases of the radius of convergence: (1)When then, the radius of convergence is 2(2)When then, the radius of convergence is greater than 2(3)When then, the radius of convergence is less than 2
In general, we conclude that the radius of convergence, is such that where . Then, we consider the interval of convergence is We consider a case of .
Suppose and are distinct values, that is, . Then, we have that
is a normalised univalent function which belongs to class with beta function as its coefficients. Hence,
If ; then, equation (9) reduces to
From equations (9) and (11), we have that and as the coefficients of (9) and (11), respectively. In addition, belongs to the class with negative coefficients so that and .
Let
Equations (9) an (13) are special case of (1) and (2), respectively.
Given , let where . Then, . But rather, we consider the case where and give the coefficients as follows:
If , the table below shows the possible coefficients of a normalised univalent function associated with the Eulerian integral of the first kind (Table 1). It is obvious from the below table that for all .
Hence, equation (9) becomes
Consider the special case of (9) with such that , we have that
In [2, 7], the authors introduced a generalised multiplier transformation as a tool to define a subclass of univalent functions. The application of the multiplier transformation was considered in [8]. One of the interests in complex analysis is that every analytic function has a power series expansion. This further encourages the use of a normalised univalent functions of the form (1); and in this work, we consider a normalised univalent function whose coefficients are in terms of the Eulerian integral of the first kind. Altunhan and Eker in [9] earlier explored subclasses of univalent functions with coefficients in Poisson distribution series while many other authors have also considered various forms of special functions, see [4, 10, 11] for examples.
2. Main Results
Theorem 1. Let the function be defined by (13). Then,
Proof. Let , then is nonnegative. That is, The sequence is a monotonic decreasing sequence such that as and by letting , we have that Since . Then, for
Theorem 2. Let the function belong to class . Then, the function defined as also belongs to class with
Proof. By definition, Since belongs to class for every and . Then, for every and . Thus, which completes the proof.
Theorem 3. Let the function be in the class and let be a real number such that . Then, the function also belongs to the class .
Proof. From the definition, we can express as follows where Thus, which completes the proof.
Theorem 4. Let be a real number such that . If , then, the function defined by (41) is univalent in where
Proof. Let be defined by (13). Then, from (41), it follows that We now need to show that in where Hence, provided From (29), we have that (48) is satisfied if which implies that as required. Thus, the function is univalent in with the best estimate
3. Application of Fractional Calculus
We now consider an application of fractional calculus earlier introduced by [12, 13] to the univalent function involving beta function as follows.
Definition 5. The fractional integral of order for a function is defined as where , is an analytic function in a simply connected region of the plane containing the origin and the multiplicity of is removed by requiring to be real when
Definition 6. The fractional integral of order for a function is defined by where , is an analytic function in a simply connected region of the plane containing the origin and the multiplicity of is removed by requiring to be real when
Definition 7. Under the hypotheses of Definition 6., the fractional derivative of order is defined by where and .
Theorem 8. Let . Then,
Proof. Using the above definition and upon simple calculation, we have that setting so that is a function of and such that . It is obvious that is a strictly monotonic decreasing sequence since for all values of . It thus follows that Thus, Hence, as required.
4. Conclusion
In this study, a certain univalent function associated with the Eulerian integral of the first kind, otherwise known as beta function, is established and some of the geometric properties of the functions discussed. The study opens up a new direction of investigation in geometric function theory (GFT).
Data Availability
No data was used.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The first author would like to acknowledge Landmark University, Nigeria.