On the Second Hankel Determinant for a New Subclass of Analytic Functions
1Department of Mathematics, M.S.K. Girls College, Bharowal (Tarn-Taran), Punjab, India
2Department of Mathematics, Guru Nanak Dev University College, Chungh (Tarn-Taran), Punjab, India
In the present investigation an upper bound of the second Hankel determinant for the functions belonging to the class R(α;A,B) is studied. The results due to various authors follow as special cases.
Keywords: analytic functions, Subordination, Schwarz function, second Hankel determinant
Journal of Mathematical Sciences and Applications, 2014 2 (1),
Received December 16, 2013; Revised January 06, 2014; Accepted February 10, 2014Copyright © 2013 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Singh, Gagandeep, and Gurcharanjit Singh. "On the Second Hankel Determinant for a New Subclass of Analytic Functions." Journal of Mathematical Sciences and Applications 2.1 (2014): 1-3.
- Singh, G. , & Singh, G. (2014). On the Second Hankel Determinant for a New Subclass of Analytic Functions. Journal of Mathematical Sciences and Applications, 2(1), 1-3.
- Singh, Gagandeep, and Gurcharanjit Singh. "On the Second Hankel Determinant for a New Subclass of Analytic Functions." Journal of Mathematical Sciences and Applications 2, no. 1 (2014): 1-3.
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Let A be the class of analytic functions of the form
in the unit disc .
Let S be the class of functions and univalent in E.
Let U be the class of Schwarz functions
which are analytic in the unit disc and satisfying the conditions and
Let f and g be two analytic functions in E. Then f is said to be subordinate to g (symbolically if there exists a Schwarz function , such that
denotes the subclass of functions in satisfying the condition
In particular, , the class studied by Murugusundramurthi and Magesh . Also , the class introduced and studied by Goel and Mehrok  and , the class of functions whose derivative has a positive real part introduced and studied by MacGregor .
In 1976, Noonan and Thomas  stated the qth Hankel determinant of for and as
where are the coefficients of various powers of z in as defined in (1.1).
For our discussion in this paper, we consider the Hankel determinant in the case of and, known as the second Hankel determinant:
2. Preliminary Results
Let P be the family of all functions p analytic in E for which and
If P, then .2.2. Lemma 2.2. [5,6]
If P, then
for some x and z satisfying and .
3. Main Result3.1. Theorem 3.1
If , then
If , then there exists a Schwarz function such that
Define the function by
Since is a Schwarz function, we see that and Define the function by
In view of the equations (3.2), (3.4) and (3.5), we have
Using (3.3) and (3.5) in (3.6), we obtain
Using Lemma 2.1 and Lemma 2.2 in (3.8), we obtain
Assume that and , using triangular inequality and , we have
is an increasing function.
Consequently, we get
Hence from (3.9), we obtain (3.1).
The result is sharp for , and
For and in Theorem 3.1, we obtain the following result due to Murugusundramurthi and Magesh .3.3. Corollary 3.1.1
If , then
For , Theorem 3.1 gives the following result.3.4. Corollary 3.1.2
If , then
Putting , and in Theorem 3.1, we obtain the following result due to Janteng et al. .3.5. Corollary 3.1.3
If , then
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