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American Journal of Public Health Research
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The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

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The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

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Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

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The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

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

Gagandeep Singh*, Gurcharanjit Singh

Department of Mathematics, M.S.K. Girls College, Bharowal (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.

Keywords: analytic functions, Subordination, Schwarz function, second Hankel determinant

Received December 16, 2013; Revised January 06, 2014; Accepted February 10, 2014

Citation: Gagandeep Singh, Gurcharanjit Singh, On the Second Hankel Determinant for a New Subclass of Analytic Functions. Journal of Mathematical Sciences and Applications, 2 (1). 1-3. January 2014. doi:10.12691/jmsa-2-1-1

1. Introduction

Let A be the class of analytic functions of the form

[picture] | (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

[picture] | (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

[picture] | (3.1)

3.2. Proof

If , then there exists a Schwarz function such that

[picture] | (3.2)

where

[picture] | (3.3)

Define the function by

[picture] | (3.4)

Since is a Schwarz function, we see that and Define the function by

[picture] | (3.5)

In view of the equations (3.2), (3.4) and (3.5), we have

Thus,

[picture] | (3.6)

Using (3.3) and (3.5) in (3.6), we obtain

[picture] | (3.7)

(3.7) yields,

[picture] | (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

[picture] | (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

[1]   | R. M. Goel and B. S. Mehrok, A subclass of univalent functions, J. Austral. Math. Soc.(Series A), 35 (1983), 1-17.
[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.
[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.
[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.
[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.
[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.
[7]   | T. H. Mac Gregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537.
[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.
[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.
[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.
[11]   | Ch. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht, 1975.
[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.
[13]   | Gagandeep Singh, Hankel determinant for a new subclass of analytic functions, Scientia Magna, 8 (4) (2012), 61-65.