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Convolution of harmonic mappings on the exterior unit disk and the generalized hypergeometric functions.

1 Introduction

Complex-valued harmonic univalent functions have recently been studied from the perspective of geometric function theory. These studies were inspired by the seminal works of Clunie and Sheil-Small [6], and also by Sheil-Small [21] on the class Sh consisting of complex-valued harmonic orientation-preserving univalent mappings / defined on the open unit disk XI, and normalized at the origin by f(0) = 0 and [f.sub.z](0) = 1. Various subclasses of [S.sub.H] have since been investigated by several authors (see for example [2,4, 8, 9,11,18,19, 22, 25]).

In [7], Hengartner and Schober investigated the family [[summation].sub.H] consisting of harmonic orientation-preserving univalent mappings f defined on [??] = {z : [absolute value of z] > 1} that map [infinity] to [infinity]. Such a mapping admits a representation of the form

f(z)=Alog [absolute value of z]+h(z)+ [bar.g(z)],

where

h(z) = [alpha]z + [[infinity].summation over(n=0)] [a.sub.n][z.sup.-n], and g(z) = [beta]z + [[infinity].summation over (n=1)] [b.sub.n] [z.sup.-n]

are analytic in [??], and [absolute value of a] > [absolute value of [beta]]. In addition, the function defined by a = [[bar.f].sub.[bar.z]]/[f.sub.z] is analytic and satisfies [absolute value of a(z)] < 1. By applying an affine transformation ([bar.[alpha]]f - [[bar.[beta]]f] - [bar.[alpha]] [a.sub.0] + [bar.[beta]][a.sub.0]/[([absolute value of [alpha]].sup.2] - [[absolute value of [beta]].sup.2], we may restrict our attention to the family [[summation].sub.H] of harmonic functions of the form

f(z) = z + Alog[absolute value of z] + [[infinity].summation over (n=1)][a.sub.n][z.sup.-n] + [[bar.[infinity].summation over (n=1)] [b.sub.n][z.sup.-n]].

The subclass with no logarithmic singularity will be denoted by [[summation]".sub.H]:= {f [member of] [[summation]'.sub.H] : A = 0}. Thus functions f [member of] [[summation]".sub.H] have the representation f = h + [bar.g], where

h(z) = z + [[infinity].summation over (n=1)[a.sub.n] [z.sup.-n] and g{z) = [[infinity]. summation over (n=1)] [b.sub.n] [z.sup.-n] (1.1)

are analytic in [??]. Several subclasses of the family EH have been studied in [1, 10,12, 20]. In [10], the class of univalent harmonic functions starlike of a certain order was considered, and sufficient coefficient conditions were obtained. In [20] a class of harmonic functions related to the analytic univalent classes of uniformly convex functions and parabolic starlike functions [17] was investigated.

Now let [sigma] be a real constant satisfying [absolute value of [sigma]] = 1, and [[PHI].sub.[sigma]] = [[phi].sub.1] + [sigma][bar.[[sigma].sub.2]], where [[phi].sub.1] and [[phi].sub.2] are two analytic functions in [??], with

[[phi].sub.1](z) = z + ([[infinity].summation over (n=0)] [A.sub.n][z.sup.-n]] and [[phi].sub.2](z) = z [[infinity].summation over (n=0)] [B.sub.n] [z.sup.-n]. (1.2)

In this paper, a new subclass of functions in [[[summation]".sub.H] defined by convolution is introduced. This subclass encompasses several classes investigated earlier, particularly those studied in [10, 20]. For that purpose, let us first recall the definition of convolution of two harmonic mappings.

If f = h + [bar.g] is given by (1.1), and[[PHI].sub.[sigma]] by (1.2), then the convolution [[PHI].sub.[sigma]] * f in [??] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With F(z) = ([[PHI].sub.[sigma]] * f)(z) and 0 [less than or equal to] [alpha] < 1, the function f is said to belong to the class [[summation].sub.H]([[PHI].sub.[sigma]], [alpha]) provided F [member of] [[summation]".sub.H] and

[partial derivative]/[partial derivative][theta] arg(f([re.sup.i[theta]])) > [alpha]

on [absolute value of z] = r for each r > 1 and 0 [less than or equal to] [theta] < 2[pi]. Specifically, the class [[summation].sub.H]([[PHI].sub.[sigma]],[alpha]) is given in the following definition:

Definition 1.1. Let [sigma] be a real constant with [absolute value of [sigma]] = 1, and 0 [less than or equal to] [alpha] < 1. Let [[PHI].sub.[sigma]](z) = [[phi].sub.1] (z) + [sigma][[phi].sub.2](z) be a given harmonic function in [??], where [[phi].sub.1] and [[phi].sub.2] are of the form (1.2). A harmonic function / = h + g where h and g are of the form (1.1), belongs to the class [[summation].sub.H]([[PHI].sub.[sigma],[alpha]) if [[PHI].sub.[sigma]] * f [member of] [[summation]".sub.H] satisfies the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)

Several subclasses of harmonic functions are special cases of the class [[summation].sub.H]([[PHI].sub.[sigma],[alpha]). Notable among these subclasses are the subclasses [[summation]*.sub.H](a) of harmonic starlike functions and [[summation].sub.KH]([alpha]) of harmonic convex functions investigated by Jahangiri [10], where

[[summation].sub.H]([[PHI].sub.1,[alpha]]) = [[summation]*.sub.H]([alpha]) and [[summation].sub.H] ([[PHI].sub.- 1,[alpha]) = [[summation].sub.KH]([alpha]) (1.4) respectively, with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the class [[summation].sub.H]([[PHI].sub.[sigma],alpha]) provides a unified treatment of various subclasses of harmonic mappings under appropriate choices of the parameter [sigma] and harmonic function [PHI].

In the next section of this paper, a necessary and sufficient convolution condition is obtained for the class [[summation]".sub.H]([PHI].sub.[sigma],[alpha]), which as application, yields a sufficient coefficient condition for the class. In Section 3, an appropriate general class of harmonic functions in [[summation]".sub.H] with negative coefficients is defined. Necessary and sufficient coefficient conditions are obtained. Growth estimates and extreme points are also determined for the class. In Section 4, starlikeness conditions of the Liu-Srivastava operator involving the generalized hypergeometric functions are investigated. Since many operators can be expressed in terms of the hypergeometric functions, the inclusion results obtained here will be useful for several other operators.

We shall require the following result:

Theorem 1.1. [12] If f of the form (1.1) satisfies the inequality

[[infinity].summation over (n=1)] n ([absolute value of [a.sub.n]] + [absolute value of [b.sub.n]]) [less than or equal to] 1, (1.5)

then f is a harmonic, orientation-preserving univalent function in [??].

2 Main Results

We now derive a convolution characterization for functions in the class [[summation].sub.H] ([[PHI].sub.[sigma],[alpha]).

Theorem 2.1. (Convolution Condition) Let f = h + [bar.g] [member of] [[summation]".sub.H], and 0 [less than or equal to] [alpha] < 1. A function f belongs to [[summation].sub.H]([[PHI].sub.[sigma],[alpha]) if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. A necessary and sufficient condition for f = h + [bar.g] to be in the class [[summation].sub.[sigma],[alpha]), with h and g of the form (1.1), is given by (1.3). The condition (1.3) holds if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.1)

By a simple algebraic manipulation, (2.1) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The latter condition, along with (1.3) for x = - 1, establishes the result for all [absolute value of x]= 1.

An application of the convolution condition in Theorem 2.1 yields a sufficient coefficient condition for harmonic functions to belong to the class [[summation].sub.H] ([[PHI].sub.[sigma],[alpha]).

Theorem 2.2. If f = h + [bar.g] of the form (1.1) and = [[PHI].sub.[sigma] = [[phi].sub.1] + [sigma][[bar.[phi]].sub.2] of the form (1.2) satisfy the coefficient inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

then f [member of] [[summation].sub.H]([[PHI].sub.[sigma],[alpha]).

Proof. The given condition shows that the coefficients of [[PHI].sub.[sigma]]* f satisfy

[[infinity].summation over (n=1)] n([absolute value of [a.sub.n]] [absolute value of [A.sub.n]] + [absolute value of [b.sub.n][absolute value of [B.sub.n])] [less than or equal to] 1.

It follows from (1.5) in Theorem 1.1 that [[PHI].sub.[sigma]] * f [member of] [[summation]".sub.H]. For h and g given by (1.1),

Theorem 2.1 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The last expression is non-negative by hypothesis, and hence by Theorem 2.1, it follows that f [member of] [[summation].sub.H] ([[PHI].sub.[sigma], [alpha]).

Remark 2.1. The coefficient bound in Theorem 2.2 can also be found in [10]. However the approach is different in this paper.

Using the relations (1.4), along with Theorem 2.2 yield the following two corollaries:

Corollary 2.1. [10] Let f = h + [bar.g] be of the form (1.1), and 0 [less than or equal to] [alpha] < 1. If

[[infinity].summation over (n=1)] [(n + [alpha])[absolute value of [a.sub.n]] + (n - [alpha])[absolute value of [b.sub.n]]] [less than or equal to] 1 - [alpha],

then f [member of] [[summation]*.sub.H] ([alpha]).

Corollary 2.2. [10] Let f = h + [bar.g] be of the form (1.1), and 0 [less than or equal to] [alpha] < 1. If

[[infinity].summation over (n=1)] [(n + [alpha])[absolute value of [a.sub.n]] + (n - [alpha])[absolute value of [b.sub.n]]] [less than or equal to] 1 - [alpha],

then f [member of] [[summation].sub.KH]([alpha]).

3 Harmonic mappings with negative coefficients

In this section, we shall devote attention to an appropriate subclass of harmonic functions with negative coefficients. Let us denote by T[[summation]".sub.H] the class consisting of functions f = h + [bar.g] [member of] [[summation]".sub.H], where

h[z) = z + [sigma] [[infinity].summation over (n=1)] [a.sub.n] [z.sup.-n], and g(z) = - [[infinity].summation over (n=1)] [b.sub.n] [z.sup.-n], [a.sub.n] [greater than or equal to] 0, [b.sub.n] [greater than or equal to] 0. (3.1)

Let [[PHI].sub.[sigma] = ([phi].sub.1] + [sigma] [bar.[[sigma].sub.2]], where

[[phi].sub.1](z) = z + [sigma] [[infinity].summation over (n=0)] [A.sub.n] [z.sup.-n], [[phi].sub.2](z)= z + [sigma] [[infinity].summation over [B.sub.n] [z.sup.-n], ([A.sub.n] [greater than or equal to] 0, [B.sub.n] [greater than or equal to]> 0), (3.2)

are given analytic functions in [??], and the real constant [sigma] satisfies [absolute value of [sigma]] = 1.

We shall use the notation

T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]]: = [[summation].sub.H]([[PHI].sub.[sigma],[alpha]) [intersection] T[[summation]".sub.H],

and for the harmonic starlike situation, we let

T[[summation]* .sub.H]([alpha]) := [[summation]*.sub.H]([alpha])[intersection]T[[summation]".sub.H].

A necessary and sufficient coefficient condition is obtained for the class T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]]).

Theorem 3.1. Let f be of the form (3.1), and 0 [less than or equal to] [alpha] < 1. The function f belongs to T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]) if and only if

[[infinity].summation over (n=1)] n + [alpha]/1 - [alpha] [a.sub.n] [A.sub.n] + [[infinity].summation over (n=1)] [b.sub.n] [B.sub.n] [less than or equal to] 1. (3.3)

Proof. If f belongs to T[[summation].sub.H] ([[PHI].sub.[sigma],[alpha]), then (1.3) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for z [member of] [??]. Letting z [right arrow] 1 + through real values yields condition (3.3). The fact that condition (3.3) is sufficient is obtained from Theorem 2.2.

From (1.4), Theorem 3.1 yields the following result:

Corollary 3.1. [10] Let f be of the form (3.1), and 0 [less than or equal to] [alpha] < 1. Then f [member of] T[summation]*.sub.H] ([alpha]) if and only if

[[infinity].summation over (n=1)] [(n + [alpha])[a.sub.n] + (n - [alpha])[b.sub.n]] [less than or equal to] 1 - [alpha].

Also f [member of] T[summation].sub.KH] ([alpha]) if and only if

Theorem 3.2. Let [[PHI].sub.[sigma]] be of the form (3.2) with [A.sub.n] [greater than or equal to] [A.sub.1] > 0, [B.sub.n] [greater than or equal to] [B.sub.1] > 0, and 1 [less than or equal to] [B.sub.1] [less than or equal to] [A.sub.1]. If f [member of] T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]), then for [absolute value of z] = r > 1,

r - 1/[B.sub.1][r.sup.-1] [less than or equal to] [absolute value of f(z)] [less than or equal to] r + 1/[B.sub.1][r.sup.-1].

Proof. First note that by assumptions,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The lower bound is obtained in a similar manner.

The lower bound is sharp with equality for f{z) = z - 1/[[sigma].sup.2][B.sub.1][[bar.z].sup.1]. The estimates given in the corollary below improve the bounds obtained by Jahangiri [10].

Corollary 3.2. If f [member of] T[[summation]*.sub.H]([alpha]) or f [member of] T[[summation].sub.KH]([alpha]), then

r - [r.sup.-1] [less than or equal to][absolute value of f(z)][less than or equal to] [r.sup.-1], \z\ = r > 1.

The class T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]]) is clearly convex. We now determine its extreme points.

Theorem 3.3. Let

[h.sub.0]{z): = z, [h.sub.n]{z): = z + [sigma](1 - [alpha])/(n + [alpha]) [A.sub.n] [z.sup.-n],

and

[g.sup.0](z):= z, [g.sub.n](z):= z - 1 - [alpha]/(n - [alpha])[B.sub.n][[bar.z].sup.-n] (n = l,2, ...).

A function f [member of] T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]) if only if f can be expressed in the form

f(z) = [[infinity].summation over (n=0)] ([[lambda].sub.n][h.sub.n](z) + [[gamma].sub.n][g.sub.n](z)),

where [[lambda].sub.n] [greater than or equal to] 0, and [summation].sup.[infinity].sub.n=0] ([[lambda].sub.n] + [[gamma].sub.n]) = 1.

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

it follows from Theorem 3.1 that f [member of] T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]]).

Conversely, if f [member of] T[[summation].sub.H]([[PHI].sub.[sigma],[alpha]), then

[a.sub.n] [less than or equal to] 1 - [alpha]/(n + [alpha]) [A.sub.n], and [b.sub.n] [less than or equal to] 1 - [alpha]/(n - [alpha])[B.sub.n]

For n [greater than or equal to] 1, set

[[lambda].sub.n] = n + [alpha]/1 - [alpha] [a.sub.n], [A.sub.n] [[gamma].sub.n] = n - [alpha]/1 - [alpha] [b.sub.n] [B.sub.n], 0 [less than or equal to] [[lambda].sub.0] [less than or equal to] 1,

and

[[gamma].sub.0] = 1 - [[lambda].sub.0] - [[infinity].summation over (n=1)] ([[lambda].sub.n] + [[gamma].sub.n]).

Then it is easily seen that [[summation].sup.[infinity].sub.n=0] ([[lambda].sub.n][h.sub.n](z) + [[gamma].sub.n][g.sub.n](z)) = f(z).

4 The Liu-Srivastava Linear Operator

As applications in this final section, we take the operator [[PHI].sub.[sigma] discussed in the earlier sections to be the Liu-Srivastava operator involving the generalized hypergeometric functions. For that purpose, first let us denote by E the class of all analytic functions f in [??] of the form

f(z) = z + [[infinity].summation over (k=0)] [a.sub.k] [z.sup.-k].

For [[alpha].sub.j] [member of] C (j = 1,2, ..., l) and [[beta].sub.k] [member of] C \ {0,-1,-2,- ...} (k = 1,2, ...m), the generalized hypergeometric function [sub.l][F.sub.m] ([[alpha].sub.1], ..., [[alpha].sub.1]; [[beta].sub.1], ..., [[beta].sub.m];z) in [??] is defined by the infinite series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(l [less than or equal to] m + 1;l,m [member of] [N.sub.o] : = {0,1,2, ...}), where [(a).sub.n] is the Pochhammer symbol given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is known [23, p.43] that the /Fm series is absolutely convergent in C if l [less than or equal to] m, and in [??] if l = m + 1. Furthermore, if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

then the [.sub.l][F.sub.m] series is absolutely convergent for [absolute value of z] = 1. Corresponding to the function z [.sub.l][F.sub.m]([[alpha].sub.1], ..., [[alpha].sub.l];[[beta].sub.1], ..., [[beta].sub.m];z), the Liu-Srivastava operator [5,15,16]

[H.sup.(l,m)]([[alpha].sub.1], ..., [[alpha].sub.l]; [[beta].sub.1] ..., [[beta].sub.m]): [??] [right arrow][??]

is defined by the Hadamard product

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For convenience, we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Special cases of the Liu-Srivastava linear operator include the Carlson-Shaffer linear operator L(a, c) := [H.sup.(2,1)] (1, a; c) (studied among others by Liu and Srivastava [14], Liu [13], and Yang [27]), the operator [D.sup.n+1] := L(n + 1,1), which is analogous to the Ruscheweyh derivative operator (investigated by Yang [26]), and the operator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(studied by Uralegaddi and Somanatha [24]).

Corresponding to f = h + [bar.g] given by (1.1), we define an operator L on f given by

L[f] = [[PHI].sub.[sigma] * f = ([[phi].sub.1] + [sigma][[bar.[sigma]].sub.2]) * (h + [bar.g]), (4.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.3)

Of course here we are assuming that none of the denominator parameters can be zero or a negative integer. A similar operator to L defined by (4.1) was recently studied in the unit disk by Ahuja et al. [3].

Theorem 4.1. Let f = h + [bar.g] [member of] [[summation]".sub.H] be of the form (1.1), where the coefficients [a.sub.n] and [b.sub.n] satisfy

[absolute value of [a.sub.n]] [less than or equal to] 1 - [alpha]/n + [alpha], and [absolute value of [b.sub.n]] [less than or equal to] 1 - [alpha]/n - [alpha], (n [greater than or equal to] 1). (4.4)

Let [[phi].sub.1] and [[phi].sub.2] of the form (4.2) satisfy

[m.summation over (j=1)] [beta].sub.j] > [l.summation over (j=1)] [absolute value of [[lambda].sub.j]], [q.summation over (j=1) [absolute value of [c.sub.j]],

where [[beta].sub.j] > 0 (j = 1, ..., m) and [d.sub.j] > 0 (j = 1 ,..., q). If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.5)

holds, then L[f][member of] [[summation]*.sub.H]([alpha]).

Proof. In view of Theorem 2.2, it suffices to show that S [less than or equal to] 1 - [alpha], where

S := [[infinity].summation over (n=1) (n + [alpha])[absolute value of an][absolute value of [A.sub.n]] + [[infinity].summation over (n=1) (n - [alpha]) [absolute value of [b.sub.n]] [absolute value of [B.sub.n]], (4.6)

where [A.sub.n] and [B.sub.n] are given by (4.3). Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

provided (4.5) holds.

Note that the hypergeometric condition (4.5) is independent of [alpha].

Example 4.1. Let l = 2 = p, m = 1 = q, [beta] > 1 + [absolute value of [lambda]], and d > 1 + [absolute value of c] in Theorem

4.1. The Gauss summation formula [23, p.30] gives

2[F.sub.1](a, b; c; 1) = [GAMMA](c) - a - b)/[GAMMA](c - a) [GAMMA](c - b), Re(c - a - b) > 0.

Using the property that [GAMMA] (z + 1) = z[GAMMA](z) and the Gauss summation formula, the condition (4.5) reduces to

[beta] - 1/[beta] - [absolute value of [lambda]]-1 - [lambda]/[beta] + d - 1/d - [absolute value of c] - 1 - [absolute value of c]/d [less than or equal to] 3.

Let M([alpha]) denote the class consisting of functions f = h + g of the form (1.1) satisfying

[[infinity].summation over (n=1) [n + [alpha]) [absolute value of [a.sub.n]] + (n - [alpha]) [absolute value of [b.sub.n]]] [less than or equal to] 1 - [alpha].

It follows from Corollary 2.1 that M([alpha]) [subset] [[summation]*.sub.H]([alpha]), and under conditions (4.5), the proof of Theorem 4.1 shows that L[M ([alpha])] [subset] M([alpha]) also holds true. In particular, with M([alpha]) = T[[summation]*.sub.H]([alpha]), the following corollary is obtained:

Corollary 4.1. Let L[f] be given by (4.1) with [sigma] = 1. Further let ([[phi].sub.1] and ([[phi].sub.2] of the form (4.2) satisfy

[m.summation over (j=1)] [[beta].sub.j] > [l.summation over (j=1)] [[lambda].sub.j], [q.summation over (j=1)] > [p.summation over (j=1)] [c.sub.j],

Where [[lambda].sub.j] [greater than or equal to] 0, [[beta].sub.j] > 0, and [c.sub.j] [greater than or equal to] 0, [d.sub.j] > 0. Then L (T[[summation]*.sub.H]([alpha])[subset] t[[summation]*.sub.H]([alpha]) if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.7)

Proof. It follows from Corollary 3.1 that the coefficients [a.sub.n] and [b.sub.n] satisfy the conditions (4.4) of Theorem 4.1. If the condition (4.7) holds true, it follows that S [greater than or equal to] 1 - [alpha], where S is given by (4.6). Corollary 3.1 now gives L[f] [member of] T[[summation]*.sub.H] ([alpha].

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* This work was supported in part by USM's Research University grant and FRGS grant.

Received by the editors July 2009.

Communicated by F. Brackx.

2000 Mathematics Subject Classification : Primary 30C45,30C55,58E20.

School of Mathematical Sciences

Universiti Sains Malaysia, 11800 USM Penang, Malaysia

email:rosihan@cs.usm.my,kgs@usm.my,sklee@cs.usm.my

Department of Mathematics

Madras Christian College, Chennai 600059, India

email:adolfmcc2003@yahoo.co.in
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Author:Ali, Rosihan M.; Stephen, B. Adolf; Subramanian, K.G.; Lee, S.K.
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:May 1, 2011
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