On the Second Hankel Determinant for a New Subclass of Analytic Functions

Gagandeep Singh, Gurcharanjit Singh

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On the Second Hankel Determinant for a New Subclass of Analytic Functions

Gagandeep Singh1,, Gurcharanjit Singh2

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

Abstract

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.

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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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 3.1

If , then

(3.1)
3.2. Proof

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].

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. [2].

3.5. Corollary 3.1.3

If , then

References

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[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.
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[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.
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[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.
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[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.
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[7]  T. H. Mac Gregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537.
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[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.
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[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.
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[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.
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[11]  Ch. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht, 1975.
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[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.
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[13]  Gagandeep Singh, Hankel determinant for a new subclass of analytic functions, Scientia Magna, 8 (4) (2012), 61-65.
In article      
 
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