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

[section]1. Introduction and preliminariesLet A be the class of analytic functions of the form

f(z) = z + [[infinity].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) 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 univalent /uni·va·lent/ (u?ni-va´lent) having a valence of one.

u·ni·va·lent

*adj.*

**1.**Having valence 1.

**2.**Having only one valence.

**3.**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 ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (2)

This class was studied by Darus and Thomas [1] and functions of this class are called [alpha]-logarithmically convex Convex

Curved, as in the shape of the outside of a circle. Usually referring to the price/required yield relationship for option-free bonds. 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 de·note

*tr.v.*

**de·not·ed**,

**de·not·ing**,

**de·notes**

**1.**To mark; indicate: a frown that denoted increasing impatience.

**2.**the subclass In programming, to add custom processing to an existing function or subroutine by hooking into the routine at a predefined point and adding additional lines of code.

**subclass**- derived class 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 determinant, a polynomial expression that is inherent in the entries of a square matrix. The size

*n*of the square matrix, as determined from the number of entries in any row or column, is called the order of the 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

*Besides the meaning discussed in this article, the*Hankel transform*may also refer to the determinant of the Hankel matrix of a sequence*.

In mathematics, the

**Hankel transform**of order ν of a function

*f*(

*r*of an integer sequence In mathematics, an

**integer sequence**is a sequence (i.e., an ordered list) of integers.

An integer sequence may be specified

*explicitly*by giving a formula for its

*n*th term, or

*implicitly*by giving a relationship between its terms. is defined and some of its properties discussed by Layman LAYMAN, eccl. law. One who is not an ecclesiastic nor a clergyman. [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

**Adj.**

**1.**

**generalised**- not biologically differentiated or adapted to a specific function or environment; "the hedgehog is a primitive and generalized mammal"

generalized

biological science, biology - the science that studies living organisms 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 lemma (lĕm`ə): see theorem.

(logic)

**lemma**- A result already proved, which is needed in the proof of some further result. 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

**Libera**may refer to:

- Libera (mythology), a Roman goddess of fertility and wife of Liber
- Libera (music), a boy choir from London
*Libera me*, a movement of the Requiem

Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. 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 logarithm (lŏg`ərĭ

*th*əm) [Gr.,=relation number], number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number. 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

*(Math.)*a function whose value increases when that of the variable increases, and decreases when the latter is diminished; also called a monotonically increasing function ltname>.

See also: Increase . 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 corollary: see theorem. 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 In mathematics, the method of

**equating the coefficients**is a way of solving a functional equation of two polynomials for a number of unknown parameters. It relies on the fact that two polynomial are identical precisely when all corresponding coefficients are equal. 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 This is a

**list of theorems**, by Wikipedia page. See also

- list of fundamental theorems
- list of lemmas
- list of conjectures
- list of inequalities
- list of mathematical proofs
- list of misnamed theorems
- Existence theorem

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 University Press**(known colloquially as CUP) is a publisher given a Royal Charter by Henry VIII in 1534, and one of the two privileged presses (the other being Oxford 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 integer: see number; number theory Sequences, 4(2001), 1-11.

[9] R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function In mathematics, a real-valued function

*f*defined on an interval (or on any convex subset of some vector space) is called

**convex**, or

**concave up**, if for any two points

*x*and

*y*in its domain

*C*and any

*t*in [0,1], we have

[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 |
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Publication: | Scientia Magna |

Date: | Sep 1, 2012 |

Words: | 2422 |

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