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Inclusion relations involving certain classes of p-valent meromorphic functions with positive coefficients based on a linear operator.

Abstract

In this paper two classes [M.sub.p]([gamma], a, c, [beta]) and [N.sub.p]([gamma], a, c, [beta], [mu]) of p-valent meromorphic functions of the form f(z)= 1 / [z.sup.p] + [[infinity].summation over (n=p+1)] [a.sub.n-p] [z.sup.n-p], n, p [member of] N involving a linear operator are defined. Coefficient inequalities and extreme points for functions belonging to these classes are obtained. Inclusion relations of these classes using [delta]-neighbourhood [N.sup.[delta].sub.p](I) p N (I) are also found. Further, using subordinate characterization inclusion relation between [M.sub.p]([gamma], a, c, [beta]) and [N.sub.p]([gamma], a, c, [beta], [mu]) is also obtained.

2000 Mathematics Subject Classification: 30C45.

Keywords and Phrases. p-valent functions, analytic functions, meromorphic functions, linear operator, subordination, neighbourhood.

1. Introduction

Let [M.sub.p] denotes the class of functions of the form:

f(z) = 1 / [z.sup.p] + [[infinity].summation over (n=p+1)] [a.sub.n-p] [z.sup.n-p] (p [member of] N, [a.sub.n-p] [less than or equal to] 0) (1.1)

which are analytic and p-valent in [U.sup.*] = {z [member of] C : 0 < [absolute value of z] < 1}.

The Hadamard product (or convolution) of f(z) given by (1.1) and g(z) given by

g(z) = 1 / [z.sup.p] + [[infinity].summation over (n=p+1)] [b.sub.n-p] [z.sup.n-p] ([b.sub.n-p] [greater than or equal to] 0)

is defined as

(f * g)(z) = 1 / [z.sup.p] + [[infinity].summation over (n=p+1)] [a.sub.n-p] [b.sub.n-p] [z.sup.n-p].

For real or complex numbers a and c (c [not equal to] 0, -1, -2, ...), we define a function [[phi].sub.p](a, c; z) as:

[[phi].sub.p](a,c;z) = 1 / [z.sup.p] + [[infinity].summation over (n=1)] [(a).sub.n] / [(c).sub.n] [z.sup.n-p] (1.2)

where [(x).sub.n] denotes the Pochhammer symbol defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is noted that

[[phi].sub.p](a, c; z) = [z.sup.-p][sub.2][F.sub.1](1, a, c; z)

where

[sub.2][F.sub.1] (1,a,c;z) = [[infinity].summation over (n=0)] [(1).sub.n] [(a).sub.n] / [(c).sub.n] [z.sup.n] / n!

Now a linear operator [L.sub.p](a, c) on [M.sub.p] using convolution is defined as:

[L.sub.p](a, c) f(z) = [[phi].sub.p](a, c; z) * f(z) (1.3)

or,

[L.sub.p] (a, c)f(z) = 1 / [z.sup.p] + [[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] [a.sub.n-p][z.sup.n-p] (p [member of] N, [a.sub.n-p] [greater than or equal to] 0). (1.4)

It is easily verified that if f(z) [member of] [M.sub.p], then

z([L.sub.p](a, c)f(z))' = a[L.sub.p](a + 1, c) f(z) - (a + p)[L.sub.p](a, c)f(z). (1.5)

A function p(z) analytic in the open unit disk U with p(0) = p is said to be in [P.sub.p]([beta], [gamma]) class if and only if

[absolute value of p(z) - p] < (p - [beta]) [absolute value of [gamma]], 0 [less than or equal to] [beta] < p and [gamma] [member of] C\{0} (1.6)

or equivalently

p(z) < p + (p - [beta]) [absolute value of [gamma]] z, z [member of] U = {z [member of] C : [absolute value of z] < 1}. (1.7)

The class [P.sub.p]([beta], [gamma]) has very close relation with various well known classes of analytic functions. For example [P.sub.1](1- b, 1) [equivalent to] P(1, b) which was defined and studied by Janowski [1] and its subordination characterization was studied by Silverman and Silvia [3] (see also [2]).

We define following sub-classes of [M.sub.p] involving linear operator [L.sub.p](a, c).

Definition 1.1: A function f(z) [member of] [M.sub.p] is said to be in [M.sub.p]([gamma], a, c, [beta]) if and only if

-z ([L.sub.p] (a,c)f(z))' / [L.sub.p] (a,c)f(z) [member of] [P.sub.p] ([beta], [gamma])

where 0 [less than or equal to] b < p and [gamma] [member of] C\{0}

or equivalently,

-z ([L.sub.p] (a,c)f(z))' / [L.sub.p] (a,c)f(z) / <p + (p - [beta]) [absolute value of [gamma]]z. (1.8)

Definition 1.2: A function f(z) [member of] [M.sub.p] is said to be in [N.sub.p]([gamma], a, c, [beta], [mu]) if and only if

{(1 - [mu])[pz.sup.p] ([L.sub.p] (a,c)f(z))) - [[mu]z.sup.p+1] ([L.sub.p] (a,c)f(z))'} [member of] [P.sub.p] ([beta], [gamma])

where p/p+1 < [mu] [less than or equal to] 1, 0 [less than or equal to] [beta] < p and [gamma] [member of] C / {0}

or equivalently,

{(1 - [mu])[pz.sup.p] ([L.sub.p] (a,c)f(z)) - [[mu]z.sup.p+1] ([L.sub.p](a,c)f(z))'} < p + (p - [beta]) [absolute value of [gamma]]z. (1.9)

Remark: If a = c, then we note that [L.sub.p](a, a)f(z) = f(z).

Thus, we denote [M.sub.p]([gamma], a, a, [beta]) = [M.sub.p]([gamma], [beta]) and [N.sub.p]([gamma], a, a, [beta], [mu]) = [N.sub.p]([gamma], [beta], [mu]).

For f(z) [member of] [M.sub.p] the neighbourhood of the function f(z) is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [delta] [greater than or equal to] 0.

In particular for I(z) = 1 / [z.sup.p], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [delta] [less than or equal to] 0.

In this paper, we study p-valent meromorphic functions and obtain coefficient inequalities and extreme points for functions belonging to [M.sub.p]([gamma],a,c,[beta]) and [N.sub.p]([alpha], a, c, [beta], [mu]) classes. Also we obtain inclusion relations of these classes involving a [delta]-neighbourhood [N.sup.[delta].sub.p] (I). Further, two new classes of [M.sub.p] are defined and inclusion relations of neighbourhoods of functions in the classes [M.sub.p]([gamma],a,c,[beta]) and [N.sub.p]([gamma], a, c, [beta], [mu]) with these new classes are found. Using subordinate characterization an inclusion relation between [M.sub.p]([gamma],a,c,[beta]) and [N.sub.p]([gamma], a, c, [beta], [mu]) is also derived.

The concept of [delta]-neighbourhood of analytic functions f(z) was introduced by Ruscheweyh [4] and Goodman [5], but for meromorphic p-valent function was studied by Liu and Srivastava [6].

In order to prove our results, we need following lemma.

Lemma 1.3: (Jack's Lemma [9]): Let w(z) be analytic in U with w(0) = 0. If [absolute value of w(z)] attains its maximum value on the circle [absolute value of z] = r < 1 at a point [z.sub.0], we can write [z.sub.0]w'([z.sub.0]) = kw([z.sub.0]) for some k [greater than or equal to] 1.

2. Coefficient Inequalities

In this section, necessary and sufficient condition for function f(z) to be in [M.sub.p]([gamma], a, c, [beta]) and [N.sub.p]([gamma], a, c, [beta], [mu]), classes are obtained. Extreme points for these classes are also found.

Theorem 2.1: A function f(z) [member of] [M.sub.p] is in [M.sub.p]([gamma], a, c, [beta]) if and only if

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] (n + (p - [beta]) [absolute value of [gamma]]) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]], (n, p [member of] N, 0 [less than or equal to] [beta] < p and [gamma] [member of] C {0}). (2.1)

Proof: Let f(z) [member of] [M.sub.p]([gamma], a, c, [beta]) then we have

[absolute value of -z ([L.sub.p] (a,c)f(z))' / [L.sub.p] (a,c)f(z) - p] [less than or equal to] (p - [beta]) [absolute value of [gamma]]

or

[absolute value of z([L.sub.p])(a,c)f(z))' + p ([L.sub.p] (a,c)f(z)) / [L.sub.p] (a,c)f(z)] [less than or equal to] (p - [beta]) [absolute value of [gamma]].

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] {n + (p - [beta]) [absolute value of [gamma]]} [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]],

which proves (2.1).

Conversely, Let (2.1) be true, then consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, f(z) [member of] [M.sub.p]([gamma], a, c, [beta]).

Corollary 2.2: A function f(z) [member of] [M.sub.p] is in [M.sub.p]([gamma], [beta]) if and only if

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

(n, p [member of] N, 0 [less than or equal to] [beta] < p and [gamma] [member of] C \ {0}).

Proof: The result holds by substituting a = c in Theorem 2.1.

Theorem 2.3: A necessary and sufficient condition for f(z) [member of] [M.sub.p] to be in [N.sub.p]([gamma], a, c, [beta], [mu]) is

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] ([[mu].sub.n] - p) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] (2.2)

where p / p + 1 < [mu] [less than or equal to] 1, 0 [less than or equal to] [beta] < p and [gamma] [member of] C \ {0}.

Proof: Let f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) then we have

[absolute value of (1 - [mu]) [pz.sup.p] ([L.sub.p] (a,c)f(z)) - [[mu]z.sup.p+1]([L.sub.p] (a,c)f(z))' - p] < (p - [beta]) [absolute value of [gamma]]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[absolute value of [[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] (p - [mu]n) [a.sub.n-p][z.sup.n]] < (p - [beta]) [absolute value of [gamma]]

or

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] ([mu]n - p) [a.sub.n-p] < (p - [beta]) [absolute value of [gamma]], (z [right arrow] 1).

Conversely, Let (2.2) holds true for f(z) [member of] [M.sub.p] then consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]). This completes the proof.

Corollary 2.4: A necessary and sufficient condition for f(z) [member of] [M.sub.p] satisfying

-[z.sup.p+1]([L.sub.p](a, c)f(z))' [member of] [P.sub.p]([beta], [gamma])

is

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] (n - p) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]]. (2.3)

Proof: The result holds by substituting [mu] = 1 in Theorem 2.3.

Putting a = c in Theorem 2.3 and Corollary 2.4, following corollaries can be obtained.

Corollary 2.5: A necessary and sufficient condition for f(z) [member of] [M.sub.p] is to be in [N.sub.p]([gamma], [beta], [mu]) is

[[infinity].summation over (n=p+1)] ([mu]n - p) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]]

where

p / p + 1 < [mu] [less than or equal to] 1, 0 [less than or equal to] [beta] < p and [gamma] [member of] C \ {0}.

Corollary 2.6: A necessary and sufficient condition for f(z) [member of] [M.sub.p] satisfying

-[z.sup.p+1]f'(z) [member of] [P.sub.p]([beta], [gamma])

is

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

Theorem 2.7: The extreme points of [M.sub.p]([gamma], a, c, [beta]) are functions given by

[f.sub.p](z) = 1 / [z.sup.p] (2.4)

and

[f.sub.n] (z) = 1/[z.sup.p] + (p - [beta]) [absolute value of [gamma]] / (n + (p - [beta]) [absolute value of [gamma]]) [(c).sub.n] / [(a).sub.n] [z.sup.n-p], (n [greater than or equal to] p + 1) (2.5)

where n, p [member of] N, 0 [less than or equal to] [beta] < p.

Corollary 2.8: A function f(z) [member of] [M.sub.p]([gamma], a, c, [beta]) if and only if f(z) may be expressed as

f(z) = [[infinity].summation over (n=p)] [[mu].sub.n] [f.sub.n](z), (2.6)

where [[mu].sub.n] [greater than or equal to] 0, [[infinity].summation over (n=p)] [[mu].sub.n] = 1 and [f.sub.p], [f.sub.p+1] ... are as defined in (2.4) and (2.5).

Theorem 2.9: The extreme points of [N.sub.p]([gamma], a, c, [beta], [mu]) are functions given by

[f.sub.p] (z) = 1 / [z.sup.p] (2.7)

and

[f.sub.n] (z) = 1 / [z.sup.p] + (p - [beta]) [absolute value of [gamma]] / ([mu]n - p) [(c).sub.n] / [(a).sub.n] [z.sup.n-p], (n [greater than or equal to] p + 1) (2.8)

where p, n [member of] N, 0 [less than or equal to] [beta] < p, [gamma] [member of]C\{0} and p / p + 1 < [mu] [less than or equal to] 1.

Corollary 2.10: A function f(z) [less than or equal to] [N.sub.p]([gamma], a, c, [beta], [mu]) if and only if f(z) may be expressed as:

f(z) = [[infinity].summation over (n=p)] [v.sub.n][f.sub.n](z) (2.9)

where [v.sub.n] [greater than or equal to] 0, [[infinity].summation over (n=p)][v.sub.n] = 1 and [f.sub.p], [f.sub.p+1], ... are defined as in (2.7) and (2.8).

Substituting a = c in Theorem 2.7 and Theorem 2.9 following corollaries can be obtained:

Corollary 2.11: The extreme points of [M.sub.p]([gamma], [beta]) are functions given by

[f.sub.p] (z) = 1 / [z.sup.p]

and

[f.sub.n](z) = 1 / [z.sup.p] + (p - [beta])[absolute value of [gamma]] / (n + (p - [beta]) [absolute value of [gamma]] [z.sup.n-p], (n [greater than or equal to] p + 1).

Corollary 2.12: The extreme points of [N.sub.p]([gamma], [beta], [mu]) are functions given by

[f.sub.p] (z) = 1 / [z.sup.p]

and

[f.sub.n](z) = 1 / [z.sup.p] + (p - [beta]) [absolute value of [gamma]] / ([mu]n - p) [z.sup.n-p], (n [greater than or equal to] p + 1).

where p / p + 1 < [mu] [less than or equal to] 1.

3. Inclusion Relation Involving Neighbourhood

In this section inclusion relations for classes [M.sub.p]([gamma],a,c,[beta]) and [N.sub.p]([gamma],a,c,[beta],[mu]) involving [delta]-neighbourhood [N.sup.[delta].sub.p] (I) are obtained.

Theorem 3.1: [M.sub.p]([gamma], a, c, [beta]) [subset] [N.sup.[delta].sub.p]) (I)

where

[delta] = (p - [beta]) [absolute value of [gamma]] (2p + 2 (p - [beta])[absolute value of [gamma]] + 1} / {p + 1 + (p - [beta]) [absolute value of [gamma]]} [(c).sub.p+1] / [(a).sub.p+1].

Proof: Let f(z) [member of] [M.sub.p]([gamma], a, c, [beta]) then by Theorem 2.1, we get

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] {n + (p - [beta]) [absolute value of [gamma]]} [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]], ([absolute value of] < 1).

This gives

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] / {p + 1 + 1 (p - [beta])[absolute value of [gamma]]}.

Also from (2.1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[[infinity].summation over (n=p+1)] (n - p) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] {2p + 2(p - [beta]) [absolute value of [gamma]] + 1} / {p + 1 + (p - [beta]) [absolute value of [gamma]]} [(c).sub.p+1] / [(a).sub.p+1] = [delta].

Therefore, f(z) [member of] [N.sup.[delta].sub.p](I). This proves the result.

Corollary 3.2: [M.sub.p]([gamma],[beta]) [subset] [N.sup.[delta].sub.p] (I)

where

[delta] = (p - [beta]) [absolute value of [gamma]] {2p + 2(p -[beta]) [absolute value of [gamma]] + 1} / {p + 1 + (p - [beta]) [absolute value of [gamma]]}.

Proof: Putting a = c in Theorem 3.1, the result is obvious.

Theorem 3.3: [N.sub.p]([gamma], a, c, [beta], [mu]) [subset] [N.sup.[delta].sub.p] (I)

where

[delta] = (p - [beta]) [absolute value of [gamma]] / ([mu]p + [mu] - p) [(c).sub.p+1] / [(a).sub.p+1] for p / p + 1 < [mu] [less than or equal to] 1, n, p [member of] N and 0 [less than or equal to] [beta] < p.

Proof: Let f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]), by Theorem 2.2, we get

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] [a.sub.n-p] ([[mu].sub.n] - p) [less than or equal to] (p - [beta]) [absolute value of [gamma]]

or

[[infinity].summation over (n=p+1)] [(a).sub.n] / [(c).sub.n] [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] / ([mu]p + [mu] - p).

Also from (2.2),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[[infinity].summation over (n=p+1)] (n - p) [a.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] / ([mu]p + [mu] - p) [(c).sub.p+1] / [(a).sub.p+1] = [delta].

Therefore, f(z) [member of] [N.sup.[delta].sub.p]) (I). This proves the theorem.

Corollary 3.4: If [delta] = (p - [beta]) [absolute value of [gamma]] [(c).sub.p+1] / [(a).sub.p+1], then [N.sub.p] ([gamma], a, c, [beta], 1) [subset] [N.sup.[delta].sub.p](I).

Proof: The result holds by substituting [mu] = 1 in Theorem 3.3.

Corollary 3.5: [N.sub.p]([gamma],[beta],[mu]) [subset] [N.sup.[delta].sub.p](I)

where [delta] = (p - [beta])[absolute value of [gamma]] / [mu]p + [mu] - p, for p / p + 1 < [mu] [less than or equal to] 1, 0 [less than or equal to] [beta] < p.

Proof: By substituting a = c in Theorem 3.3, the result can be easily proved.

Corollary 3.6: If [delta] = (p - [beta]) [absolute value of [gamma]], then [N.sub.p]([gamma],[beta],1) [subset] [N.sup.[delta].sub.p] (I).

Proof: The result holds by substituting [mu] = 1 in Corollary 3.5.

4. Classes [M.sup.[alpha].sub.p] ([gamma],a,c,[beta]) and [N.sup.[alpha].sub.p] ([gamma], a, c, [beta], [mu])

In this section two new classes are defined and inclusion relations of [delta]-neighbourhood of functions in the classes [M.sub.p]([gamma], a, c, [beta]) and [N.sub.p]([gamma], a, c, [beta], mu]) involving these new classes are obtained.

Definition 4.1: A function f(z) [member of] [M.sub.p] is said to be in [M.sup.[alpha].sub.p] ([gamma], c, a, [beta]) if there exists a function g(z) [member of] [M.sub.p]([gamma], a, c, [beta]) such that

[absolute value of f(z) / g(z) - 1] < 1 - [alpha], z [member of] U, 0 [less than or equal to] [alpha] < 1. (4.1)

Definition 4.2: A function f(z) [member of] [M.sub.p] is said to be in [N.sup.[alpha].sub.p] ([gamma], a, c, [beta], [mu]) if there exists a function g(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) satisfying (4.1).

Theorem 4.3: For g(z) [member of] [M.sub.p]([gamma], a, c, [beta])

[N.sup.[delta].sub.p] (g) [subset] [M.sup.[alpha].sub.p] ([gamma], a, c, [beta])

where

[alpha] = 1 - [delta] {p + 1 + (p - [beta]) [absolute value of [gamma]]} [(a).sub.p+1 / {p + 1 + (p - [beta]) [absolute value of [gamma]]} [(a).sub.p+1] -(p - [beta]) [absolute value or [gamma]] [(c).sub.p+1]. (4.2)

Proof: Let f(z) [member of] [N.sup.[delta].sub.p] (g), then [[infinity].summation over (n=p+1)] (n - p) [absolute value of [a.sub.n-p] - [b.sub.n-p]] [less than or equal to] [delta] which implies

[[infinity].summation over (n=p+1)] [absolute value of [a.sub.n-p] - [b.sub.n-p]] [less than or equal to] [delta]. (4.3)

Again, as g(z) [member of] [M.sub.p]([gamma], a, c, [beta])

[[infinity].summation over (n=p+1)] [less than or equal to] (p - [beta]) [absolute value of gamma]] / {p + 1 + (p - [beta]) [absolute value of [gamma]]} [(c).sub.p+1] / [(a).sub.p+1]. (4.4)

Now using (4.3) and (4.4),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [alpha] is given by (4.2). This completes the proof.

Theorem 4.4: If g(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) and

[alpha] = 1 - [delta] ([mu]p + [mu] - p [(a).sub.p+1] / ([mu]p + [mu] - p) [(a).sub.p+1] - (p - [beta]) [absolute value of [gamma]] [(c).sub.p+1] (4.5)

where p / 1 + p < [mu] [less than or equal to] 1, 0 [less than or equal to] [beta] < p, [gamma] [member of] C\{0} then, [N.sup.[delta].sub.p] (g) [subset] [N.sup.[alpha].sub.p] ([gamma], a, c, [beta], [mu]).

Proof: Let f(z) [member of] [N.sup.[delta].sub.p](g), then

[[infinity].summation over (n=p+1)] [absolute value of [a.sub.n-p] - [b.sub.n-p]] [less than or equal to] [delta]. (4.6)

Again, as g(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) we have

[[infinity].summation over (n=p+1)] [b.sub.n-p] [less than or equal to] (p - [beta]) [absolute value of [gamma]] [(c).sub.p+1] / ([mu]p + [mu] - p) [(a).sub.p+1]. (4.7)

Now using (4.6) and (4.7) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [alpha] is given by (4.5). This completes the proof.

5. The Class [N.sub.p] ([gamma], a, c, [beta], [mu]) Using Subordinate Characterization

To prove the theorem, following Lemma of Miller and Mocanu [7] is needed: Lemma 5.1[7]: Let q(z) be univalent in the unit disk U = {z : [absolute value of z] < 1} and [theta] and [phi] be analytic in a domain [OMEGA] containing q(U) with [phi](w) [not equal to] 0 when w [member of] q(U). Set Q(z) = zq'(z)[phi](q(z)), h(z) = [theta](q(z)) + Q(z). Suppose that either h(z) is convex or Q(z) is starlike univalent in U. In addition, assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If p(z) is analytic in U, with p(0) = q(0), p(U) [subset or equal to] [OMEGA] and

[theta] (p(z)) + zp'(z)[phi](p(z)) < [theta] (q(z)) + zq'(z)[phi](q(z)) (5.1)

then p(z) < q(z).

Using Lemma 5.1, we prove the following lemma:

Lemma 5.2: Let q(z) be univalent and q(z) [not equal to] 0 in U and

(i) zq'(z)/q(z) is starlike univalent in U

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

p.p(z) - [mu]zp'(z) < p.q(z) - [mu]zq'(z) (5.2)

then p(z) < q(z).

Proof: Define the function [theta] and [phi] by

[theta](w) = pw and [phi](w) = -[mu],

then [theta] and [phi] are analytic in C\{0} and [phi](w) [not equal to] 0.

Define the function Q(z) and h(z) by

Q(z) = zq'(z) [phi](q(z)) = -[mu]zq'(z)

and h(z) = [theta](q(z)) + Q(z) = pq(z) -[mu]zq'(z).

In view of the hypothesis, the functions Q(z) and h(z) satisfy the condition of Lemma 5.1. So using Lemma 5.1, it is proved that

p(z) < q(z).

Theorem 5.3: If f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) then

[pz.sup.p][L.sub.p](a,c)f(z) < p + (p / p - [mu]) (p - [beta]) [absolute value of gamma]] z, (p / p + 1 < [mu] < 1). (5.3)

Proof: Define the functions p(z) and q(z) by

p(z) = [z.sup.p] [L.sub.p](a, c)f(z), (z [member of] U)

and

q(z) = 1 + (p - [beta]) / p - [mu] [absolute value of] z.

Then a simple computation shows that

p.p(z) - [mu]zp'(z) = (1 - [mu])p [z.sup.p]([L.sub.p](a, c)f(z)) - [[mu]z.sup.p+1]([L.sub.p](a, c)f(z))'

and

p.q(z) - [mu]zq'(z) = p + (p - [beta]) [absolute value of [gamma]] z.

Hence, using (1.9) and Lemma 5.2 the result is proved.

Corollary 5.4: If f(z) [member of] [N.sub.p]([gamma], [beta], [mu]), then

[pz.sup.p]f(z) < p + (p / p - [mu]) (p - [beta]) [absolute value of [gamma]] z, (p / p + 1 < [mu] < 1

Proof: The result holds by substituting a = c in Theorem 5.3.

Theorem 5.5: For p / p + 1 < [[mu].sub.1] < [mu] < 1, 0 [less than or equal to] [beta] < p and [gamma] [member of] C\{0}, let [[gamma].sub.1] = (p - [[mu].sub.1] / p - [mu]) [gamma]

[N.sub.p] ([gamma], a, c, [beta], [mu]) [subset] [N.sub.p] ([[gamma].sub.1], a, c, [beta], [[mu].sub.1])

Proof: Suppose that f(z) [member of] [N.sub.p] ([gamma], a, c, [beta], [mu]). Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On using (1.9) and (5.3), it follows that

f(z) [member of] [N.sub.p]([[gamma].sub.1], a, c, [beta], [[mu].sub.1])

where

[[gamma].sub.1] = (p - [[mu].sub.1] / p -[mu]) [gamma].

Corollary 5.6: [N.sub.p]([gamma], [beta], [mu]) [subset] [N.sub.p]([[gamma].sub.1], [beta], [[mu].sub.1]).

where p / p + 1 < [[mu].sub.1] < [mu] < 1, [[gamma].sub.1] = (p - [[mu].sub.1] / p - [mu]) [gamma], 0 [less than or equal to] [beta] < p and [gamma] [member of]C\{0}.

To prove our next theorem, we need the following lemma which is due to Miller and Mocanu [8].

Lemma 5.7[8]: Let q(z) = 1 + [q.sub.n][z.sup.n] +.... (n [greater than or equal to] 1) be analytic in U and let h(z) be convex univalent in U with h(0) = 1. If

q(z) + 1 / x zq' (z) < h (z)

for c > 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 5.8: Let

[[gamma].sub.1] = a[gamma] / (a) + p + 1),

then

[N.sub.p]([gamma], a + 1, c, [beta], [mu]) [subset] [N.sub.p]([[gamma].sub.1], a, c, [beta], [[mu].sub.1]).

Proof: For f(z) [member of] [M.sub.p], suppose that f(z) [member of] [N.sub.p]([gamma], a + 1, c, [beta], [mu]). Set

[p.sub.1](z) = (1 - [mu])[z.sup.p] [L.sub.p](a + 1, c)f(z) - [[mu]z.sup.p+1] ([L.sub.p] (a + 1, c)f(z))' / p

and

[p.sup.2](z) = (1 - [mu])[z.sup.p] [L.sub.p](a, c)f(z) - [[mu]z.sup.p+1] ([L.sub.p](a,c)f(z)' / p.

Now using (1.5) we get

[p.sub.1](z) = [p.sub.2](z) + z / a [p'.sub.2] (z).

Thus, Lemma 5.7 concludes that f(z) [member of] [N.sub.p]([[gamma].sub.1], a, c, [beta], [mu]) where

[[gamma].sub.1] = a[gamma] / (a + p + 1).

Theorem 5.9: Let

[absolute value of [gamma]] < (p - [mu] / p - [beta]).

If f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) then

[N.sub.p]([gamma], a, c, [beta], [mu]) [subset] [M.sub.p]([[gamma].sub.1], a, c, [beta])

where

[[gamma].sub.1] = (2p - [mu]) [gamma] / [mu][(p - [mu]) - (p - [beta]) [absolute value of [gamma]]].

Proof: Since f(z) [member of] [N.sub.p]([gamma], a, c, [beta], [mu]) we have

[absolute value of (1 - [mu])[z.sup.p][L.sub.p](a,c)f(z) - [[mu]z.sup.p+1] / p ([L.sub.p](a,c)f(z))'-1] < (p - [beta]) [absolute value of [gamma]] / p.

Also Theorem 5.3 gives

[absolute value of [z.sup.p][L.sub.p](a,c)f(z) - 1] < (p - [beta] / p - [mu]) [absolute value of [gamma]].

Now for

[[gamma].sub.1] = (2p - [mu]) [gamma] / [mu][(p - [mu]) - (p - [beta]) [absolute value of [gamma]]],

define w(z) as

1 + (p - [beta] / p) [absolute value of [[gamma].sub.1]] w(z) -z([L.sub.p] (a,c)f(z)' / [L.sub.p] (a,c)f(z)

where w(z) is analytic in U and w(0) = 0. To conclude the proof, it suffices to show that [absolute value of w(z)] < 1 in U. If it is not the case, then by Jack's lemma, there exists a point [z.sub.0] [member of] U such that [absolute value of w([z.sub.0])] = 1 and [z.sub.0]w'([z.sub.0]) = kw([z.sub.0]), k [greater than or equal to] 1. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Which is a contradiction to the hypothesis. So [absolute value of w(z)] < 1 in U. Thus, f(z) [member of] [M.sub.p]([[gamma].sub.1], a, c, [beta]). This completes the proof.

References

[1] W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math, 23(1970), 159-177.

[2] J.M. Jahangiri, H. Silverman and E.M. Silvia, Inclusion relations between classes of functions defined by subordination, J. Math. Anal. App., 151(1990), 318-329.

[3] H. Silverman and E.M. Silvia, Subclasses of starlike functions subordinate to convex functions, Canad. J. Math., 37(1985), 48-61.

[4] S. Rusheweyh, Neighbourhoods of univalent functions, Proc. Amer. Math. Soc. 81(1981), 521-527.

[5] A.W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8(1957), 598-601.

[6] Jimn-Lin-Liu and H.M. Srivastava, A linear operator and associated families of meromorphically multivalent functions, J. Math. Anal. And Appl. 259(2001), 566-601.

[7] S.S. Miller and P.T. Mocanu, Differential subordinations: Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics (No. 225), Marcel Dekker, New York and Basel, 2000.

[8] S.S. Miller and P.T. Mocanu, Differential subordination and univalent functions, Michigan Math J., 28(1981), 157-171.

[9] I.S. Jack, Functions starlike and convex of order a, J. London Math. Soc. 3(1971), 469-474.

Poonam Sharma * and Deepaly Chowdhary

Department of Mathematics & Astronomy, University of Lucknow, Lucknow--226007 INDIA E-mail: poonambaba@yahoo.com
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