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Estimate of second Hankel determinant for certain classes of analytic functions.

[section]1. Introduction and preliminaries

Let A be the class of analytic functions of the form

f(z) = z + [[infinity].summation over k=2] [a.sub.k][z.sup.k] (1)

in the unit disc E = {z :[absolute value of z] < 1}. Let S be the class of functions f (z) [member of] A and univalent in E. Let [M.sup.[alpha]](0 [less than or equal to] [alpha] [less than or equal to] 1) be the class of functions which satisfy the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

This class was studied by Darus and Thomas [1] and functions of this class are called [alpha]-logarithmically convex functions. Obviously [M.sup.0] = S*, the class of starlike functions and [M.sup.1] = K, the class of convex functions.

In the sequel, we assume that (0 [less than or equal to] [alpha] [less than or equal to] 1) and z [member of] E.

[C.sup.*([alpha]).sub.s] denote the subclass of functions f(z) [member of] A and satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The following observations are obvious:

(i) [C.sup.*(0).sub.s] [equivalent to] [S.sup.*.sub.s], the class of starlike functions with respect to symmetric points introduced by Sakaguchi [14].

(ii) [C.sup.*(1).sub.s] [equivalent to] [K.sub.s], the class of convex functions with respect to symmetric points introduced by Das and Singh [2].

[C.sup.[alpha].sub.s] be the subclass of functions f (z) [member of] A and satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where

g(z) = z + [[infinity].summation over k=2] [b.sub.k][z.sub.k] [member of] [S.sup.*.sub.s]. (5)

In particular

(i) [C.sup.0.sub.s] = [C.sub.s], the class of close-to-convex functions with respect to symmetric points introduced by Das and Singh [2].

(ii) [C.sup.1.sub.s] [equivalent to] [C'.sub.S].

Let [C.sup.[alpha].sub.1(s)] be the subclass of functions f (z) [member of] A and satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where

h(z) = z + [[infinity].summation over k=2] [d.sub.k][z.sup.k] [member of] [K.sub.s]. (7)

We have the following observations:

(i) [C.sup.0.sub.1(s)] [equivalent to] [C.sub.1(s)].

(ii) [C.sup.1.sub.1(s)] [equivalent to] [C'.sub.1(s)].

In 1976, Noonan and Thomas [11] stated the qth Hankel determinant for q [greater than or equal to] 1 and n [greater than or equal to] 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This determinant has also been considered by several authors. For example, Noor [12] determined the rate of growth of [H.sub.q] (n) as [right arrow] [infinity] for functions given by Eq. (1) with bounded boundary. Ehrenborg t3l studied the Hankel determinant of exponential polynomials and the Hankel transform of an integer sequence is defined and some of its properties discussed by Layman [8]. Also Hankel determinant was studied by various authors including Hayman [5] and Pommerenke [13]. Easily, one can observe that the Fekete-Szego functional is [H.sub.2] (1). Fekete and Szego [4] then further generalised the estimate of [absolute value of [a.sub.3] - [mu][a.sup.2.sub.2]] where [mu] is real and f [member of] S. For our discussion in this paper, we consider the Hankel determinant in the case of q = 2 and n = 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper, we seek upper bound of the functional [absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] for functions belonging to the above defined classes.

[section]2. Main result

Let P be the family of all functions p analytic in E for which Re(p(z)) > 0 and

p(z) = 1 + [p.sub.1]z + [p.sub.2][z.sup.2] + ... (8)

for z [member of] E.

Lemma 2.1. If p [member of] P, then [absolute value of [absolute value of [p.sub.k]] [less than or equal to] 2 (k = 1, 2, 3, ...).

This result is due to Pommerenke [13].

Lemma 2.2. If p [member of] P, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some x and z satisfying [absolute value of x] [less than or equal to] 1, [absolute value of z] [less than or equal to] 1 and [p.sub.1] [member of] [0, 2].

This result was proved by Libera and Zlotkiewiez [9,10].

Theorem 2.1. If f [member of] [M.sup[alpha]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Proof. As f [member of] [M.sup.[alpha]], so from (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

On taking logarithm on both sides of (10), we get,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

An easy calculation yields,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

On substituting (12), (13) and (14) in (11), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

On equating the coefficients of z, [z.sup.2] and [z.sup.3] in (15), we obtain

[a.sub.2] = [p.sub.1]/1 + [alpha], (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Using (16), (17) and (18), it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where C([alpha]) = 144(1 + 3[alpha])[(1 + 2[alpha]).sup.2][(1 + [alpha]).sup.4].

Using Lemma 2.1 and Lemma 2.2 in (19), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume that [p.sub.1] = p and p [member of] [0, 2], using triangular inequality and [absolute value of z] [less than or equal to] 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [delta] = [absolute value of x] [less than or equal to] 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an increasing function. Therefore MaxF([delta]) = F(1). Consequently

[absolute value of [a.sub.2][a.sub.4]- [a.sup.2.sub.3]] [less than or equal to] 1/C([alpha])G(p), (20)

where G(p) = F(1).

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

A([alpha]) = [alpha](-4 + 263[alpha] + 603[[alpha].sup.2] + 253[[alpha].sup.3] + 37[[alpha].sup.4])

and

B([alpha]) = 24[alpha](11 + 36[alpha] + 38[[alpha].sup.2] + 12[[alpha].sup.3] - [[alpha].sup.4]).

Now

G'(p) = -4A([alpha])[p.sup.3] + 2B([alpha])p

and

G"(p) = -12A([alpha])[p.sup.2] + 2B([alpha])p

G'(p) = 0 gives

p[2A([alpha])[p.sup.2] - B([alpha])] = 0

G'(p) is negative at [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So MaxG(p) = G(p'). Hence from (20), we obtain (9).

The result is sharp for [p.sub.1] = p', [p.sub.2] = [p.sup.2.sub.1] - 2 and [p.sub.3] = [p.sub.1] ([p.sup.2.sub.1] - 3).

For [alpha] = 0 and [alpha] = 1 respectively, we obtain the following results due to Janteng et al. [6]

Corollary 2.1. If f (z) [member of] [S.sup.*], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1.

Corollary 2.2. If f(z) [member of] K, then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/8.

Theorem 2.2. If f [member of] [C.sup.*([alpha]).sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/[(1+2[alpha]).sup.2] (21)

Proof. Since f [member of] [C.sup.*([alpha]).sub.s], so from (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

On taking logarithm on both sides of (22), we get,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

After an easy calculation, we obtain,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

On substituting (24), (25) and (14) in (23), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

On equating coefficients of z, [z.sup.2] and [z.sup.3] in (26), we obtain

[a.sub.2] = [p.sub.1]/2(1 + [alpha]), (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Using (27), (28) and (29), it yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

where C([alpha]) = 96(1 + 3[alpha])[(1 + 2[alpha]).sup.2][(1 + [alpha]).sup.4].

Using Lemma 2.1 and Lemma 2.2 in (30), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Assume that [p.sub.1] = p and p [member of] [0, 2], using triangular inequality and [absolute value of z] [less than or equal to] 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/C([alpha]) F([delta]).

where [delta] = [absolute value of x] [less than or equal to] 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an increasing function. Therefore MaxF([delta]) = F(1).

Consequently

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/C([alpha)] G(p), (31)

where G(p) = F(1).

So

G(p) = A([alpha])[p.sup.4] - B([alpha])[p.sup.2] + 96(1 + 3[alpha])[(1 + [alpha]).sup.4],

where

A([alpha]) = [alpha](5 + 20[alpha] + 33[[alpha].sup.2] + 28[[alpha].sup.3] + 10[[alpha].sup.4])

and

B([alpha]) = 24[(1 + [alpha]).sup.2](1 + 6[alpha] + 7[[alpha].sup.2] + 4[[alpha].sup.3]).

Now

G'(p) =4A([alpha])[p.sup.3] - 2B([alpha])p

and

G"(p) = 12A([alpha])[p.sup.2] - 2B([alpha]).

G (p) = 0 gives

2p[2A([alpha])[p.sup.2] - B([alpha])] = 0.

Clearly G(p) attains its maximum value at p = 0. So MaxG(p) = G(0) = 96(1 + 3[alpha])[(1 + [alpha]).sup.4]. Hence from (31), we obtain (21).

The result is sharp for [p.sub.1] = 0, [p.sub.2] = -2 and [p.sub.3] = 0.

For [alpha] = 0 and [alpha] = 1 respectively, we obtain the following results due to Janteng et al. [7].

Corollary 2.3. If f (z) [member of] [S.sub.*.sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1.

Corollary 2.4. If f (z) [member of] [K.sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/9.

On the same lines, we can easily prove the following theorems:

Theorem 2.3. If f [member of] [C.sup.[alpha].sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] [(3 + 2[alpha]).sup.2/9[(1 + 2[alpha]).sup.2].

The result is sharp for [p.sub.1] = 0, [p.sub.2] = - 2 and [p.sub.3] = 0.

For [alpha] = 0 and [alpha] =1 respectively, we obtain the following results:

Corollary 2.5. If f (z) [member of] [C.sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1.

Corollary 2.6. If f (z) [member of] [C'.sub.s], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 25/81.

Theorem 2.4. If f [member of] [C.sup.[alpha].sub.1(s)], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] [(7 + 2[alpha]).sup.2]/81[(1 + 2[alpha]).sup.2]

The result is sharp for [p.sub.1] = 0, [p.sub.2] = -2 and [p.sub.3] = 0. For [alpha] = 0 and [alpha] = 1 respectively, we obtain the following results:

Corollary 2.7. If f (z) [member of] [C.sub.1(s)], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 49/81.

Corollary 2.8. If f (z) [member of] [C'.sub.1(s)], then

[absolute value of [a.sub.2][a.sub.4] - [a.sup.2.sub.3]] [less than or equal to] 1/9.

References

[1] M. Darus and D. K. Thomas, a-Logarithmically convex functions, Ind. J. Pure Appl. Math., 29(1998), No. 10, 1049-1059.

[2] R. N. Das and P. Singh, On subclasses of schlicht mappings, Indian J. Pure Appl. Math., 8(1977), 864-872.

[3] R. Ehrenborg, The Hankel determinant of exponential polynomials, American Mathematical Monthly, 107(2000), 557-560.

[4] M. Fekete and G. Szego, Eine Bemerkung iiber ungerade schlichte Funktionen, J.London Math. Soc., 8(1933), 85-89.

[5] W. K. Hayman, Multivalent functions, Cambridge Tracts in Math. and Math. Phys., No. 48, Cambridge University Press, Cambridge, 1958.

[6] Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1(2007), No. 13, 619-625.

[7] 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(2006), No. 1, 37-43.

[8] J. W. Layman, The Hankel transform and some of its properties, J. of Integer Sequences, 4(2001), 1-11.

[9] 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.

[10] 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.

[11] J. W. Noonan and D. K. Thomas, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc., 223(1976), No. 2, 337-346.

[12] K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl., 28(1983), No. 8, 731-739.

[13] Ch. Pommerenke, Univalent functions, Gottingen: Vandenhoeck and Ruprecht., 1975.

[14] K. Sakaguchi, On a certain Univalent mapping, J. Math. Soc. Japan, 11(1959), 72-80.

B. S. Mehrok and Gagandeep Singh

Department of Mathematics, DIPS College(Co-Ed.), Dhilwan(Kapurthala), Punjab, India

E-mail: kamboj.gagandeep@yahoo.in
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Author:Mehrok, B.S.; Singh, Gagandeep
Publication:Scientia Magna
Date:Sep 1, 2012
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