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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 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 nth 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
 and Zlotkiewiez [9,10].

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`ərĭthəm) [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
, 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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