Abstract
Let denote the open unit disk and let denote the class of normalized univalent functions which are analytic in . Let be the class of concave functions , which have the condition that the opening angle of at infinity is less than or equal to , . In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class . And we define a class , , which is a subclass of and we find the set of variabilities for the functional for . This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in . We also give a characterization for functions in in terms of Hadamard product.
1. Introduction
Let denote the class of functions analytic in the unit disk . We denote the class of locally univalent functions by . Let denote the class of functions with normalization and let be the class of functions in that are univalent in . Also we define the subclass of convex functions whenever is a convex domain.
A function is said to belong to the family if satisfies the following conditions:(i)is analytic in with the standard normalization . In addition, it satisfies .(ii) maps conformally onto a set whose complement with respect to is convex.(iii)The opening angle of at is less than or equal to , .
The class is referred to as the class of concave univalent functions. We note that for , , the closed set is convex and unbounded. We observe that contains the classes , .
Avkhadiev and Wirths [1] found the analytic characterization for functions in , : if and only if For , the pre-Schwarzian derivative is defined by and we define the norm of by It is well known that for and for . In [2], Bhowmik et al. obtained the estimate of the pre-Schwarzian norm for functions as the following: For more investigation of concave functions, we may refer to [3–7].
We say that is subordinate to in , written as , if and only if for some Schwarz functions , , and , . If is univalent in , then the subordination is equivalent to and .
By using the subordination, we define a subclass of concave functions as follows.
Definition 1. Let and be real numbers such that . The function belongs to the class if satisfies the following:
Note that .
Let , , and be complex numbers with . We define the Gaussian hypergeometric function by where is Pochhammer symbol defined, in terms of Gamma function , by We note that the Gaussian hypergeometric function satisfies the hypergeometric differential equation
The Gaussian hypergeometric function has been studied extensively by various authors [8–13]. In particular, univalency, close-to-convexity, starlikeness, convexity, and various other properties associated with these hypergeometric functions were investigated based on the conditions of , , and in [14–17].
If and given bythen Hadamard product (or convolution) of and is defined (as usual) by
In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class . And we find the set of variabilities for the functional and as a consequence of this we derive upper and lower bounds for the pre-Schwarzian norm , for functions in . And we give a representation formula in terms of Hadamard product for functions in .
2. A Sufficient Condition for Functions to Be in
In this section, we investigate a sufficient condition for the Gaussian hypergeometric functions to be in the class . The proof of our result in this section is based upon the following lemmas.
Lemma 2 (see [18], Miller and Mocanu, p. 35). Let be a set in the complex plane and let be a complex number such that . Suppose that the function satisfies the condition for all real , and all . If the function defined by is analytic in and if then in .
Lemma 3 (see [18], Miller and Mocanu, p. 239). If , , and are real and satisfythen .
Theorem 4. Let and , , and be real and satisfy (12) andin , whereThen,
Proof. Let Then, satisfies , , and From the hypergeometric differential equation (7), we haveFrom condition (12) and Lemma 3, we have in . If we setthen is analytic in and satisfies . Furthermore, we haveHence, we getIf we use this substitution in (19), thenDifferentiating (24), we haveSubstituting (23) in (25), we haveAnd this equation leads us to the following first order differential equation:where Since inequality (13) implies in , we can rewrite (27) in the form whereA bilinear transformation , with and , has the property that , for . From the condition and (13), we havein andin . Now, we let and define a function by Then, (27) becomes By using (14), (31), and (32), we obtainfor all and all real , with . By Lemma 2, we have in , which shows that .
Example 5. If we take , , , and , then we can easily check that conditions (12) and (13) are satisfied. Furthermore, where(see Figure 1). Hence, belongs to , as shown in Figure 2.
Example 6. If we take , , , and , then we can easily check that conditions (12) and (13) are satisfied. Furthermore, where (see Figure 3). Hence, belongs to , as shown in Figure 4.
3. The Pre-Schwarzian Norm Estimate for Functions in
Now, for , we find the exact set of variabilities for the functional , which gives both sharp upper and lower bounds for the pre-Schwarzian norm .
Theorem 7. Let and be fixed and let . Then, the set of variabilities of the functional is the closed disk with center and radiusThe points on the boundary of this disk are attained if and only if is one of the functions , where
Proof. We use the characterization (4) for functions in and the representation where is a unimodular bounded analytic function. It follows thatBy a routine computation, one recognizes that Hence, the condition is equivalent toThis proves the first part of the assertion in the theorem. The second part follows from the fact that if and only if , , and that the solution of the differential equation (4) in this case is given by . The relation between boundary points of the above circle and the extremal function becomes clear from the identity This completes the proof of the theorem.
Remark 8. If we put and in Theorem 7, then we can obtain the result in Bhowmik et al. ([2], Theorem 2.5).
From the inequality (48), we can have the following corollary.
Corollary 9. Let , and . Then,The equality holds in lower estimate for the function and in upper estimate for the function which are described in Theorem 7.
As a consequence of Theorem 7, we can obtain a distortion theorem for the functions in .
Theorem 10. Let and . Then, for each , one haswith . For each , , equality occurs if and only if , where .
Remark 11. If we put and in Theorem 10, then we can obtain the distortion theorem for the functions in which is a result from Bhowmik et al. ([2], Theorem 2.8).
Finally, we present a characterization for functions in the class , in view of the Hadamard product.
Theorem 12. Let and . Then, if and only iffor all and for all with . Equivalently, this holds if and only if where with
Remark 13. If we put and in Theorem 12, then we can obtain the convolution result for the functions in which is a result from Bhowmik et al. ([2], Theorem 3.1).
Conflict of Interests
The authors declare that they have no competing interests.