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

If P, then .

If P, then

for some x and z satisfying and .

3. Main Result

If , then

(3.1)

If , 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].

If , then

For , Theorem 3.1 gives the following result.

If , then

Putting , and in Theorem 3.1, we obtain the following result due to Janteng et al. ^[2].

If , then

References

[1]	R. M. Goel and B. S. Mehrok, A subclass of univalent functions, J. Austral. Math. Soc.(Series A), 35 (1983), 1-17.
	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), 1-5, 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), 619-625.
	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), 37-43.
	In article
[5]	R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85 (1982), 225-230.
	In article		CrossRef
[6]	R. J. Libera and E-J. Zlotkiewiez, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251-257.
	In article
[7]	T. H. Mac Gregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537.
	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), 85-94.
	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), 85-89.
	In article
[10]	J. W. Noonan and D. K. Thomas, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc., 223 (2) (1976), 337-346.
	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), 67-76.
	In article
[13]	Gagandeep Singh, Hankel determinant for a new subclass of analytic functions, Scientia Magna, 8 (4) (2012), 61-65.
	In article