Keywords: analytic functions, Subordination, Schwarz function, second Hankel determinant
Journal of Mathematical Sciences and Applications, 2014 2 (1),
pp 13.
DOI: 10.12691/jmsa211
Received December 16, 2013; Revised January 06, 2014; Accepted February 10, 2014
Copyright: © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
Let A be the class of analytic functions of the form
 (1.1) 
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
 (1.2) 
In particular, , the class studied by Murugusundramurthi and Magesh ^{[9]}. Also , the class introduced and studied by Goel and Mehrok ^{[1]} and , the class of functions whose derivative has a positive real part introduced and studied by MacGregor ^{[7]}.
In 1976, Noonan and Thomas ^{[10]} 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:
and obtain an upper bound to for Earlier Janteng et al. ^{[2, 3, 4]}, Mehrok and Singh ^{[8]}, Singh ^{[12, 13]} have obtained sharp upper bounds of for different classes of analytic functions.
2. Preliminary Results
Let P be the family of all functions p analytic in E for which and
2.1. Lemma 2.1. [11]If P, then .
2.2. Lemma 2.2. [5,6]If P, then
for some x and z satisfying and .
3. Main Result
3.1. Theorem 3.1If , then
 (3.1) 
3.2. ProofIf , then there exists a Schwarz function such that
 (3.2) 
where
 (3.3) 
Define the function by
 (3.4) 
Since is a Schwarz function, we see that and Define the function by
 (3.5) 
In view of the equations (3.2), (3.4) and (3.5), we have
Thus,
 (3.6) 
Using (3.3) and (3.5) in (3.6), we obtain
 (3.7) 
(3.7) yields,
 (3.8) 
where .
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.
Therefore
Consequently, we get
 (3.9) 
where
So
where
Now
and
So
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 ^{[9]}.
3.3. Corollary 3.1.1If , then
For , Theorem 3.1 gives the following result.
3.4. Corollary 3.1.2If , then
Putting , and in Theorem 3.1, we obtain the following result due to Janteng et al. ^{[2]}.
3.5. Corollary 3.1.3If , then
References
[1]  R. M. Goel and B. S. Mehrok, A subclass of univalent functions, J. Austral. Math. Soc.(Series A), 35 (1983), 117. 
 In article  CrossRef 

[2]  Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Ineq. Pure Appl. Math., 7 (2) (2006), 15, Art. 50. 
 In article  

[3]  Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1 (13) (2007), 619625. 
 In article  

[4]  Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for functions starlike and convex with respect to symmetric points, J. Quality Measurement and Anal., 2 (1) (2006), 3743. 
 In article  

[5]  R. J. Libera and EJ. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85 (1982), 225230. 
 In article  CrossRef 

[6]  R. J. Libera and EJ. Zlotkiewiez, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251257. 
 In article  

[7]  T. H. Mac Gregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532537. 
 In article  CrossRef 

[8]  B. S. Mehrok and Gagandeep Singh, Estimate of second Hankel determinant for certain classes of analytic functions, Scientia Magna, 8 (3) (2012), 8594. 
 In article  

[9]  G. Murugusundramurthi and N. Magesh, Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bull. Math. Anal. and Appl., 1 (3) (2009), 8589. 
 In article  

[10]  J. W. Noonan and D. K. Thomas, On the second Hankel determinant of a really mean pvalent functions, Trans. Amer. Math. Soc., 223 (2) (1976), 337346. 
 In article  

[11]  Ch. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht, 1975. 
 In article  

[12]  Gagandeep Singh, Hankel determinant for new subclasses of analytic functions with respect to symmetric points, Int. J. of Modern Math. Sci., 5 (2) (2013), 6776. 
 In article  

[13]  Gagandeep Singh, Hankel determinant for a new subclass of analytic functions, Scientia Magna, 8 (4) (2012), 6165. 
 In article  
