# Subordination and superordination for multivalent functions defined by linear operators.

1. Introduction

Let H be the class of functions analytic in the unit disk D := {z [member of] C : [absolute value of z] < 1} and H[a, n] be the subclass of H, which contains functions of the form f (z) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [A.sub.p] denote the class of all analytic functions of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and let [A.sub.1] := A. For two functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the Hadamard product (or convolution) of f and g is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For two analytic functions f and g, we say that f is subordinate to g or g superordinate to f, denoted by f < g, if there is a Schwarz function w with [absolute value of w(z)] [less than or equal to] [absolute value of z] such that f(z) = g(w(z)). If g is univalent, then f < g if and only if f(0) = g(0) and f(D) [subset or equal to] g(D). The class [TAU]([alpha]) is defined to be the class of all functions f [member of] A satisfying Re(f(z)/z) > [alpha], 0 [less than or equal to] [alpha] < 1, z [member of] D and let [TAU] := [TAU](0). For an analytic function [phi] with [phi](0) = 1, let S*([phi]) denote the class of all f [member of] A satisfying zf'(z)/f (z) < [phi](z). Several special choices of [phi] reduce to well-known classes. For -1 [less than or equal to] B < A [less than or equal to] 1, S* [A, B] := S* ((1 + Az)/(1 + Bz)) is the Janowski starlike functions [13] (see [28]) and S*[1 - 2 [alpha], -1] is the class S*([alpha]) of starlike functions of order [alpha] and S* := S*(0) is the class of starlike functions. For 0 < [eta] [less than or equal to] 1, the class S* (((1 + z)/[(1 - z).sup.[eta]]) is the class SS*([eta]) of strongly starlike function of order [eta]. For [eta] > 0, the class S* ([(1 + z).sup.[eta]]) is the class SL([eta]); the class SL := SL(1/2) was introduced by Sokol and Stankiewicz [35] and studied recently by Ali et al. [1].

For [[alpha].sub.j] [member of] C (j = 1, 2, ...,l) and [[beta].sub.j] [member of] C \{0, -1, -2, ...} (j = 1, 2, ... m), the Dziok-Srivastava operator [11, 36] [H.sup.l'm.sub.p] [[[alpha].sub.1]] = [H.sup.(l'm).sub.p] ([[alpha].sub.1], ..., [[alpha].sub.l]; [[beta].sub.1], ..., [[beta].sub.m]) is defined by

[H.sup.(l'm).sub.p]([[alpha].sub.1], ..., [[alpha].sub.l]; [[beta].sub.1], ...,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [(a).sup.n] is the Pochhammer symbol defined by [(a).sup.n] := [GAMMA](a + n)/[GAMMA](a). Several interesting properties of the classes defined by DziokSrivastava operator or its various particular cases including the Hohlov operator [12], the Carlson-Shaffer operator (cf. [7, 18]), the Ruscheweyh derivatives [31], the generalized Bernardi-Libera-Livingston integral operator (cf. [4, 16, 19]) and the Srivastava-Owa fractional derivative operators (cf. [25, 26]), rests on the following identity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

The multiplier transformation [I.sub.p](r, [lambda]) on [A.sub.p], introduced by Sivaprasad Kumar et al. [33] and investigated in [2, 3, 34], defined by the following infinite series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

satisfies the identity:

z([I.sub.p](r, [lambda])f(z))' = (p + [lambda])[I.sub.p](r + 1, [lambda])f(z) - [lambda][I.sub.p](r, [lambda])f (z). (1.4)

The operator [I.sub.p](r, [lambda]) is closely related to the Salagean derivative operators [32]. The operator [I.sup.r.[lambda]] := [I.sub.1](r, [lambda]) was studied by Cho and Srivastava [9] and Cho and Kim [10]. The operator [I.sub.r] := [I.sub.1](r, 1) was studied by Uralegaddi and Somanatha [37]. Several other operators investigated recently also satisfies a relation similar to the relations (1.2) and (1.4). Notable among them are the operators introduced by Al-Kharasani and Al-Areefi [3] which includes the operators defined in [15], [23] and [22] as well as the Jung-Kim-Srivastava operator [14] and its p-valent analogue of Liu [17].

In the following definition, all these operators investigated one by one are unified.

Definition 1.1. Let [O.sub.p] be the class of all linear operators [L.sup.a.sub.p] defined on [A.sub.p] satisfying

z[[L.sup.a.sub.p]f(z)]' = [[alpha].sub.a][L.sup.a+1.sub.p]f(z) - ([[alpha].sub.a] - p)[L.sup.a.sub.p]f(z).

One can also consider operators satisfying z[[L.sup.b.sub.p]f(z)]' = [[alpha].sub.b][L.sup.b-1.sub.p]f(z) --([[alpha].sub.b]-p) [L.sup.b.sub.p](z) but their properties are very similar to the operators in the above definition. In the following sections, several subordination and superordination theorems as well as corresponding sandwich theorems are proved. A related integral transform is also discussed. Further several sufficient conditions for functions to belong to the classes S, S* ([alpha]), SS*([eta]) and SL are investigated. Our results are motivated by recent results of Miller and Mocanu [21] on second order differential superordinations. Their results were later used extensively by Bulboaca [5, 6] to investigate superordination-preserving integral operators as well as by several others [2, 3, 11, 29, 30, 33, 34, 36].

We need the following:

Definition 1.2. [21, Definition 2, p.817] Denote by Q, the set of all functions f(z) that are analytic and injective on [bar.D]--E(f), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and are such that f'([zeta]) [not equal to] 0 for [zeta] [member of] [partial derivative]D - E (f).

Lemma 1.1 (cf. Miller and Mocanu[20, Theorem 3.4h, p.132]). Let [psi](z) be univalent in the unit disk D and let [??] and [psi] be analytic in a domain D [contains] [psi](D) with [psi](w) [not equal to] 0, when w [member of] [phi](D). Set

Q(z) := z[phi]'(z)[phi]([psi](z)), h(z) := [??]([psi](z)) + Q(z).

Suppose that

1. Q(z) is starlike univalent in D and

2. Re (zh'(z)/Q(z)) > 0 for z [member of] D.

If q(z) is analytic in D, with q(0) = [psi](0), q(D) [subset] D and

[??](q(z)) + zq'(z)[psi](q(z)) < [??]([psi](z)) + z[psi]'(z)[phi]([psi](z)), (1.5)

then q(z) < [psi](z) and [psi](z) is the best dominant.

Lemma 1.2. [6, Corollary 3.2, p.289] Let [psi](z) be univalent in the unit disk D and [??] and [psi] be analytic in a domain D containing [psi](D). Suppose that

1. Re [[??]'([psi](z))/[phi]([psi](z))] > 0 for z [member of] D,

2. Q(z) := z[psi]'(z)[phi]([psi](z)) is starlike univalent in D.

If q(z) [member of] H[[psi](0), 1] [intersection] Q, with q(D) [subset or equal to] D, and [??](q(z)) + zq'(z)[phi](q(z)) is univalent in D, then

[??]([psi](z)) + z[psi]'(z)[phi]([psi](z)) < [??](q(z)) + zq'(z)[phi](q(z)), (1.6)

implies [psi](z) < q(z) and [psi](z) is the best subordinant.

2. Subordination, Superordination and Sandwich Results

For functions f, F [member of] [A.sub.p], let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the powers are principal one, [mu] and v are real numbers such that they do not assume the value zero simultaneously.

Theorem 2.1. Let [psi] be convex univalent in D with [psi](0) = 1 and f [member of] [A.sub.p]. Let [[alpha].sub.a+1] [not equal to] 0, Re[[[alpha].sub.a+1][mu] - [[alpha].sub.a]v] [greater than or equal to] 0. Assume that [chi] and [PHI] are respectively defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

and

[PHI](z) := [[OMEGA].sup.a.sub.L,[mu],v] (f(z))[[??].sub.L](z), (2.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

1. If [PHI](z) [??] [chi](z), then

[[OMEGA].sup.a.sub.L,[mu],v](f(z)) < [psi](z)

and [psi](z) is the best dominant.

2. If [chi](z) < [PHI](z),

0 [not equal to][[OMEGA].sup.a.sub.L,[mu],v](f(z)) [member of] H[1,1][intersection] Q and [PHI](z) is univalent in D, (2.3)

then

[psi](z) [??] [[OMEGA].sup.a.sub.L,[mu],v](f(z)) and [psi](z) is the best subordinant.

Proof. Define the function q by

q(z) := [[OMEGA].sup.a.sub.L,[mu],v](f(z)), (2.4)

where the branch of q(z) is so chosen such that q(0) = 1. Then q(z) is analytic in D. By a simple computation, we find from (2.4) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

By making use of the identity

Z([L.sup.a.sub.p]f(z))' = [[alpha].sub.a][L.sup.a+1.sub.p]f(z) - ([[alpha].sub.a] - p)[L.sup.a.sub.p]f(z), (2.6)

in (2.5), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

In view of (2.7), the subordination [PHI](z) < [chi](z) becomes

([[alpha].sub.a+1][mu] - [[alpha].sub.a]v)q(z) + zq'(z) [??] ([[alpha].sub.a+1][mu] -[[alpha].sub.a]v)[psi](z) + z[psi]'(z)

and this can be written as (1.5), by defining

[??](w) := ([[alpha].sub.a+1][mu] - [[alpha].sub.a]v)w and [phi](w) := 1.

Note that [psi](w) [not equal to] 0 and [??](w), [phi](w) are analytic in C - {0}. Set

Q(z) := z[psi]'(z)

h(z) := [??]([psi](z)) + Q(z) = ([[alpha].sub.a+1][mu] -[[alpha].sub.a]v)[psi](z) + z[psi]'(z).

In light of the hypothesis of our Theorem 2.1, we see that Q(z) is starlike and

Re(zh'(z)/Q(z)) = Re ([[alpha].sub.a+1][mu] - [[alpha].sub.a]v + 1 + z[psi]"(z)/[psi]'(z)) > 0.

By an application of Lemma 1.1, we obtain that q(z) [??] [psi](z) or

[[OMEGA].sup.a.sub.L,[mu],v] (f(z)) [??] [psi](z).

The second half of Theorem 2.1 follows by a similar application of Lemma 1.2.

Using Theorem 2.1, we obtain the following "sandwich result".

Corollary 2.1. Let [[psi].sub.j] (j = 1, 2) be convex univalent in D with [[psi].sub.j](0) = 1. Assume that Re [[[alpha].sub.a+1][mu] - [[alpha].sub.a]v] [greater than or equal to] 0 and [PHI] be as defined in (2.2). Further assume that

[[chi].sub.J](z) := 1/[[alpha].sub.a+1][([[alpha].sub.a+1][mu] - [[alpha].sub.a]v)[[psi].sub.j](z)+[z[phi]'.sub.j](z)].

If (2.3) holds and [[chi].sub.1](z) < [PHI](z) [??] [[chi].sub.2], then

[[psi].sub.1](z) < [[OMEGA].sup.a.sub.L,[mu],v](f(z)) < [[psi].sub.2](z).

Theorem 2.2. Let [psi] be convex univalent in D with [psi](0) = 1 and [[alpha].sub.a] be a complex number. Assume that Re([micro][[alpha].sub.a+1] - v[[alpha].sub.a]) [greater than or equal to] 0 and f [member of] [A.sub.p]. Define the functions F, [chi] and [PSI] respectively by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

[chi](z) := ([micro][[alpha].sub.a+1] - v[[alpha].sub.a])[psi])(z) + z[psi]'(z) (2.9)

and

[PSI](z) := [[OMEGA].sup.a.sub.L,[mu],v](F(z))[[micro][[alpha].sub.a+1][[OMEGA].sup.a.sub.L,1,0] (f(z), F (z)), F(z)) - v[[alpha].sub.a][[OMEGA].sup.a.sub.L, 0,-1] (f(z), F(z))]. (2.10)

1. If [PSI](z) < [chi](z), then

[[OMEGA].sup.a.sub.L,[mu],v](F(z)) < [psi](z)

and [psi](z) is the best dominant.

2. If [chi](z) < [PSI](z),

0 [not equal to] [[OMEGA].sup.a.sub.L,[mu],v] (F(z)) [member of] H[1,1] [intersection] Q and [PSI](z) is univalent in D, (2.11)

then

[psi](z) < [[OMEGA].sup.a.sub.L,[mu],v](F(z))

and [psi](z) is the best subordinant.

Proof. From the definition of F, we obtain that

[[alpha].sub.a]f(z) = ([[alpha].sub.a] - p)F(z) + zF'(z). (2.12)

By convoluting (2.12) with [L.sub.a](z), where

[L.sup.a.sub.p](f(z))= [L.sub.a](z) * f(z)

and using the fact that z(f * g)'(z) = f(z) * zg'(z), we obtain

[[alpha].sub.a][L.sup.a.sub.p](f(z)) = ([[alpha].sub.a] - p)[L.sup.a.sub.p](F(z)) + z([L.sup.a.sub.p](F(z)))'. (2.13)

Define the function q by

q(z) := [[OMEGA].sup.a.sub.L,[mu],v](F(z)), (2.14)

where the branch of q(z) is so chosen such that q(0) = 1. Clearly q(z) is analytic in D. Using (2.13) and (2.14), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

Using(2.15), the subordination [PSI](z) < [chi](z) becomes

([micro][[alpha].sub.a+1] - v[[alpha].sub.a])q(z) + zq'(z) [??] ([micro][[alpha].sub.a+1] - v[[alpha].sub.a])[psi](z) + z[psi]'(z)

and this can be written as (1.5), by defining

[??](w) := ([micro][[alpha].sub.a+1] - v[[alpha].sub.a])[psi](z) and [phi](w) := 1.

Note that [phi](w) [not equal to] 0 and [??](w), [phi](w) are analytic in C - {0}. Set

Q(z) := z[psi]'(z)

h(z) := [??]([psi](z)) + Q(z) = ([micro][[alpha].sub.a+1] - v[[alpha].sub.a])[psi](z) + z[psi]'(z).

In light of the assumption of our Theorem 2.2, we see that Q(z) is starlike and

Re(zh'(z)/Q(z)) = Re([micro][[alpha].sub.a+1] - v[[alpha].sub.a] + 1 z[psi]"(z)/[psi]'(z)) > 0.

An application of Lemma 1.1, gives q(z) < [psi](z) or [[OMEGA].sup.a.sub.L,[mu],v](F(z)) < [psi](z).

By an application of Lemma 1.2 the proof of the second half of Theorem 2.2 follows at once.

As a consequence of Theorem 2.2, we obtain the following "sandwich result".

Corollary 2.2. Let [[psi].sub.j] (j = 1, 2) be convex univalent in D with [[psi].sub.j](0) = 1 and [[alpha].sub.a] be a complex number. Further assume that Re([micro][[alpha].sub.a+1] - v[[alpha].sub.a]) [greater than or equal to] 0 and [PSI] be as defined in (2.10). If (2.11) holds and [[chi].sub.1](z) < [PSI](z) < [[chi].sub.2](z), then

[[psi].sub.1](z) < [[OMEGA].sup.a.sub.L,[mu],v] (F(z)) < [[psi].sub.2](z),

where

[[chi].sub.j](z) := ([micro][[alpha].sub.a+1] - v[[alpha].sub.a] [[psi].sub.j] (z) + z[psi'].sub.j(z) (j = 1,2)

and F is defined by (2.8).

Theorem 2.3. Let [phi] be analytic in D with [phi](0) = 1 and [[alpha].sub.a] is independent of a. If f [member of] [A.sub.p], then

[[OMEGA].sup.a.sub.L,[mu],v](f(z)) [??] [phi](z) [??][[OMEGA].sup.a+1.sub.L,[mu],v] (F(z)) < [phi](z).

Further

[phi](z) [??][OMEGA].sup.a.sub.L,[mu],v](f(z)) [??] [phi](z) < [[OMEGA].sup.a+1.sub.L,[mu],v](F(z)),

where F is defined by (2.8).

Proof. Using the following identity

Z[[L.sup.a.sub.p](z))]' = [[alpha].sub.a][L.sup.a+1.sub.p](f(z)) - ([[alpha].sub.a] - p)[L.sup.a.sub.p](f(z))

in (2.13), we get

[L.sup.a.sub.p](f(z)) = [L.sup.a+1.sub.p](F(z)). (2.16)

Since [[alpha].sub.a] is independent of a, [alpha.sub.a+1] = [[alpha].sub.a], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

Therefore, from (2.16) and (2.17), we have

[OMEGA].sup.a+1.sub.L,[mu],v](F(z)) = [[OMEGA].sup.a.sub.L,[mu],v](f(z))

and hence the result follows at once. ?

Now we will use Theorem 2.3 to state the following "sandwich result".

Corollary 2.3. Let f [member of] [A.sub.p] and [[alpha].sub.a] is independent of a. Let [[phi].sub.i] (i = 1, 2) be analytic in D with [[phi].sub.i](0) = 1 and F is defined by (2.8). Then

[[phi].sub.i](z) < [[OMEGA].sup.a.sub.L,[mu],v](f(z)) < [[phi].sub.2](z)

if and only if

[[phi].sub.1](z) < [[OMEGA].sup.a+1.sub.L,[mu],v](F(z)) < [[phi].sub.2](z).

3. Applications

We begin with some interesting applications of subordination part of Theorem 2.1 for the case when L = H, the Dziok Srivastava Operator. Note that the subordination part of Theorem 2.1 holds even if we assume

Re {1 + z[psi]"(z)/[psi]'(z)} > max{0,Re[[[alpha].sub.1](v - [mu]) - [mu]]}

instead of "[psi](z) is convex and Re [[[alpha].sub.1]([mu] - v) + [mu]] [greater than or equal to] 0" and leads to the following corollary to the first part of Theorem 2.1 by taking [psi](z) = (1 + Az)/(1 + Bz).

Corollary 3.1. Let -1 < B < A [less than or equal to] 1 and Re(u - vB) [greater than or equal to] |v - [bar.u]B| where u = [[alpha].sub.1] ([mu] - v) + [mu] + 1 and v = [[[alpha].sub.1]([mu] - v) + [mu] - 1]B. If f [member of] [A.sub.p] satisfies the subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[[OMEGA].sup.[alpha]1.sub.H,[mu],v](f(z)) < [1 + Az/1 + Bz]

and (1 + Az)/(1 + Bz) is the best dominant.

Proof. Let

[phi](z) = [1 + Az/1 + Bz] (-1 < B < A [less than or equal to] 1), (3.1)

then clearly [phi](z) is univalent and [psi](0) = 1. Upon logarithmic differentiation of [phi] given by (3.1), we obtain that

z[psi](z) = (A - B)Z/[(1 + Bz).sup.2]. (3.2)

Another differentiation of (3.2), yields

1 + z[psi]"(z)/[psi]'(z) = [1 - Bz/1 + Bz].

If z = [re.sup.i[theta]], 0 [less than or equal to] r < 1, then we have

Re(1 + z[psi]"(z)/[psi]'(z)) = [1 - [B.sup.2][r.sup.2]/1 + [B.sup.2][r.sup.2] + 2Br cos[theta]] [greater than or equal to] 0.

Hence [psi](z) is convex in D. Also it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where u = [[alpha].sub.1]([mu] - v) + [mu] + 1 and v = [[[alpha].sub.1]([mu] - v) + [mu] - 1] B. The function w(z) = [u + vz/1 + Bz] maps D into the disk

[absolute value of w - [[bar.u] - [bar.v]B/1 - [B.sup.2]] [less than or equal to] [absolute value of [v- [bar.u]b|/1 - [B.sup.2]]].

Which implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided

Re([bar.u] - [bar.v]B) [greater than or equal to] [absolute value of v - [bar.u]B]

or

Re(u - vB) [greater than or equal to] [absolute value of v - [bar.u]B].

Thus the result follows at once by an application of the first part of Theorem 2.1.

Corollary 3.2. Let 0 [less than or equal to] [alpha] < 1 and Re([[alpha].sub.1]([mu] - v) + [mu]) [greater than or equal to] 0. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Let

[psi](z) = [1 + (1 - 2[alpha])z/1 - z] (0 [less than or equal to] [alpha] < 1),

then obviously [psi](z) is univalent and [psi](0) = 1. By a simple calculation, we have

1 + z[psi]"(z)/[psi]'(z) = [1 + z/1 - z],

which clearly indicates that [psi](z) is convex. If we assume [beta] = [[alpha].sub.1]([mu] - v) + [mu] then by hypothesis we have Re [beta] [greater than or equal to] 0. So if we take

w(z) = [beta] + [1 + z/1 - z] = [(1 + [beta]) + (1 - [beta])z/1 - z],

then w(z) maps the unit disc D on to Re w > Re [beta] [greater than or equal to] 0. The result now follows by an application of the subordination part of Theorem 2.1.

Note that if p =1,l = m + 1 and [[alpha].sub.i+1] = [[beta].sub.i](i = 1,2, ..., m), then [H.sub.1][1]f(z) = f(z), [H.sub.1][2]f(z) = zf'(z) and [H.sub.1][3]f(z) = 1/2[z.sup.2] f"(z) + zf'(z). Putting [alpha] = 1, p = 1, l = m + 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ..., m) in Corollary 3.2, we obtain the following.

Corollary 3.3. Let 0 [less than or equal to] [alpha] < 1 and 2[micro] [greater than equal to] v. If f [member of] A and satisfies

Re ([(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](2+ zf"(z)/f'(z)) - v zf'(z)/f(z))) > [2(2[micro] - v)[alpha] - (1 - [alpha])]/2,

then

Re([(f'(z)).sup.[mu]] [(z/f(z)).sup.v)]) > [alpha].

Proof. From Corollary 3.2, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We now investigate the image of h(D). Assuming a = 1 - 2[alpha] and b = 2[micro] - v, we have

h(z) = [b + (1 + a - b + ab)z - ab[z.sup.2]/[(1 - z).sup.2]]

where h(0) = b and h(-1) = [2b(1 - a) - (1 + a)]/4. The boundary curve of the image of h(D) is given by h([e.sup.1[theta]]) = u([theta]) + iv([theta]), -[pi] < [theta] < [pi], where

u([theta]) = [(1 + a - b + ab) + (1 - a)b cos [theta]/2(cos[theta] - 1)] and v([theta]) = [(1 + a)b sin [theta]/2(1 - cos[theta])].

By eliminating [theta], we obtain the equation of the boundary curve as

[v.sup.2] = -[b.sup.2](1 + a) (u - [2b(1 - a) - (a + 1)]/4). (3.3)

Obviously (3.3) represents a parabola opening towards the left, with the vertex at the point ([2b(1 - a) - (a + 1)]/4, 0) and negative real axis as its axis. Hence h(D) is the exterior of the parabola (3.3) which includes the right half plane

u > [2b(1 - a) - (a + 1)]/4.

Hence the result follows at once.

Setting [mu] = 0 and v = -1 in Corollary 3.3, we obtain the following result.

Example 3.1. Let 0 [less than or equal to] [alpha] < 1. If f [member of] A and Re f'(z) > 3[alpha]-1/2, then f [member of] T([alpha]).

Remark 3.1. The above Example 3.1 reduces to [24, Theorem 2] when [alpha] = 1/3.

If we take [psi](z) = [((1 + z)/(1 - z)).sup.[eta]] with 0 < [eta] [less than or equal to] 1 in Theorem 2.1 for the case L = H, the Dziok Srivastava operator, then clearly [psi](z) is convex in D and consequently corresponding to the subordination part of the Theorem 2.1, we have the following.

Corollary 3.4. Let 0 < [eta] < 1, [[alpha].sub.1] [not equal to] -1 and Re([[alpha].sub.1]([mu] - v) + [mu]) [greater than or equal to] 0. If f [member of] [A.sub.p] and satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[[OMEGA].sup.[alpha]1.sub.H,[mu],v](f(z)) < [([1 + z/1 - z]).sup.[eta]]

and [((1 + z)/(1 - z)).sup.[eta]] is the best dominant.

By taking p =1,l = m + 1, [[alpha].sub.1] = 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ... m), in the above Corollary 3.4, we have the following:

Corollary 3.5. Let 0 < [eta] [less than or equal to] 1 and 2[micro] [greater than or equal to] v. If f [member of] A and satisfies

[absolute value of arg{[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](2 + zf"(z)/f'(z)) - v zf'(z)/f(z))}] < [delta][pi]/2,

then

[absolute value of arg{[(f'(z)).sup.[mu]] [(z/f(z)).sup.v]}], [eta][pi]/2

where

[delta] = [eta] + 1 + 2/[pi] arctan 2[micro] - v/[eta].

Proof. In view of the Corollary 3.4, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

implies

[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] < ([1 + z/1 - z]).sup.[eta]]

or

[absolute value of arg{[(f'(z)).sup.[mu]] [(z/f(z)).sup.v]}] < [eta][pi]/2 (z [member of] D)

Now we need to find the minimum value of arg h(D). Let z = [e.sup.i[theta]]. Since h(D) is symmetrical about the real axis, we shall restrict ourself to 0 < [theta] < [pi]. Setting t = cot [theta]/2, we have t [greater than or equal to] 0 and for z = it - 1 / it + 1, we arrive at

h([e.sup.i[theta]]) = (it)[[eta].sup.-1] (2[mu] - v)it - [[eta](1 + [t.sup.2]]) / 2]

= (it)[[eta].sup.-1] G(t),

where

G(t) = [(2[mu] - v)it - [[eta](1 + [t.sup.-2]]) / 2].

Let G(t) = U(t) + iV(t), where U(t) = - [[eta](1 + [t.sup.2]])/2 and V(t) = (2[mu] - v)t, there arises two cases namely 2[mu] > v and 2[mu] = v. If 2[mu] > v, then a calculation shows that [min.sub.t[greater than or equal to]0] arg G(t) occurs at t = 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If 2[mu] = v, then arg G(t) = [pi] and [min.sub.[absolute value of z]<1] arg h(z) = ([eta] + 1) [pi]/2. Thus for 2[mu] [greater than or equal to] v, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof of the corollary.

We now enlist a few applications of Theorem 2.1 for the operator L = H, the Dziok Srivastava operator, by taking [psi](z) = [square root of 1 + z] as dominant. Obviously [psi](z) is a convex function in the open unit disk D with [psi](0) = 1. The subordination part of Theorem 2.1, leads to the following result.

Corollary 3.6. Let [[alpha].sub.1] [not equal to] -1 and Re[[[alpha].sub.1] ([mu] - v)+ [mu]] [greater than or equal to] 0. If f [member of] [A.sub.p] and satisfies the subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [square root of 1 + z] is the best dominant.

By taking p = 1, l = m +1, [[alpha].sub.1] = 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ... m) in Corollary 3.6, we obtain the following result.

Corollary 3.7. Let 2[mu] [greater than or equal to] v. If f [member of] A and satisfies the subordination

[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](2 + zf" (z) / f'(z)) - vzf'(z) / f(z)) < (2[mu] - v) [square root of 1 + z] + z/2[square root of 1 + z],

then

[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] < [square root of 1 + z]

and [square root of 1 + z] is the best dominant.

We obtain the following example from Corollary 3.7.

Example 3.2. If f [member of] A and satisfies

[absolute value of zf' (z) / f(z) (2 + zf"(z)/f'(z) - zf'(z)/f(z))] < [square root of 1.22] [approximately equal to] 1.10,

then f [member of] SL.

Proof. Putting [mu] = v = 1 in Corollary 3.7, we have

zf' (z) / f(z) (2 + zf"(z)/f'(z) - zf'(z)/f(z)) < [square root of 1 + z] + z/2[square root of 1 + z] =: h(z), implies

zf'(z)/f(z) < [square root of 1 + z].

The dominant h(z) can be written as

h(z) = 3z + 2 / 2 [square root of 1 + z].

Writing h([e.sup.i[theta]]) = u([theta]) + iv([theta]), -[pi] < [theta] < [pi], we have

u[theta] = [3 cos (3 [theta] / 4) + 2 cos ([theta]/4) / 2 [square root of 2 cos ([theta]/2)]]

and

v[theta] = [3 sin (3 [theta] / 4) + 2 sin ([theta]/4) / 2 [square root of 2 cos ([theta]/2)]]

A simple calculation gives

[u.sup.2]([theta]) + [v.sup.2]([theta]) = 13 + 12 cos [theta] / 8 cos ([theta]/2) =: k ([theta]).

A computation shows that k([theta]) has minimum at [theta] = arccos ([square root of 1/24]) and k([theta]) [greater than or equal to] [square root of 3/2] [approximately equal to] 1.22. Since h(0) = 1 and h(-1) = -[infinity], by a computation we come to know that the image of h(D) is the interior of the domain bounded by parabola opening towards left which contains the interior of the circle [u.sup.2] + [v.sup.2] = 1.22. Hence the result follows at once.

We now give some interesting applications of Theorem 2.2 for the case L = H. Note that if we replace the statement "[psi](z) is convex in the open unit disc D and Re [([mu] - v)[[alpha].sub.1] + [mu]] [greater than or equal to] 0" by

Re (1 + z[psi]" (z) / [psi]'(z)) > max{0, Re [(v - [mu]) [[alpha].sub.1] - [mu]]}

in the hypothesis of Theorem 2.2 still the subordination part of the result holds so we obtain the following corollary as a straight forward consequence to the first part of Theorem 2.2 by taking [psi](z) = (1 + (1 - 2[alpha])z)/(1 - z), 0 [less than or equal to] [alpha] < 1.

Corollary 3.8. Let 0 [less than or equal to] [alpha] < 1 and Re[([mu] - v)[[alpha].sub.1] + [mu]] [greater than or equal to] 0. If f [member of] [A.sub.p], F as defined in (2.8) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and (1 + (1 - 2[alpha])z)/(1 - z) is the best dominant.

Putting p =1, l = m + 1, [[alpha].sub.1] = 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ...m) in Corollary 3.8, we obtain the following result:

Corollary 3.9. Let 0 [less than or equal to] [alpha] [less than or equal to] 1 and 2[mu] [greater than or equal to] v. If f [member of] A, F as defined in (2.8) and

Re{(F '(z)).sup.[mu]] [(z / F(z)).sup.v] (2[mu] f'(z) / F'(z) - v f(z) / F(z))} < [2(2[mu] - v) [alpha] - (1 - [alpha])] / 2,

then

Re [[(F'(z)).sup.[mu]] [(z/F(z)).sup.v] > [alpha].

Proof. From Corollary 3.8, we see that

[(F '(z)).sup.[mu]] [(z / F(z)).sup.v] (2[mu] f'(z) / F'(z) - v f(z) / F(z)) < (2[mu] - v) [1 + (1 - 2[alpha])z / 1 - z] + [2(1 - [alpha])z/[(1 - z).sup.2]] =: h(z) (3.4)

implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A calculation shows that k([theta]) attains its maximum at [theta] = [pi] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the result follows at once.

By taking [psi](z) = [((1+z)/(1 - z)).sup.[eta]] in the subordination part of Theorem 2.2 for the case L = H, the Dzoik Srivastava operator, we have the following result.

Corollary 3.10. Let 0 < [eta] [less than or equal to] 1 and Re[([mu] - v)[[alpha].sub.1] + [mu]] [greater than or equal to] 0. If f [member of] [A.sub.p], F as defined in (2.8) and satisfies the subordination

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and ((1 + z)/[(1 - z)).sup.[eta]] is the best dominant.

By putting p = 1, l = m + 1, [[alpha].sub.1] = 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ... m) in the above Corollary 3.10, we obtain the following result.

Corollary 3.11. Let 0 < [eta] [less than or equal to] 1 and 2[mu] [greater than or equal to] v. If f [member of] A, F as defined in (2.8) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The proof of the above Corollary 3.11 is similar to that of the Corollary 3.5 hence it is skipped here.

Taking the dominant [psi](z) = [square root of 1 + z], which is a convex function in the open unit disc D, in the subordination part of Theorem 2.2, we have the following corollary for the Dzoik Srivastava operator H = L.

Corollary 3.12. Let 0 < [eta] [less than or equal to] 1 and Re[[[alpha].sub.1] ([mu] - v) + [mu]] [greater than or equal to] 0. If f [member of] [A.sub.p], F as defined in (2.8) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [square root of 1 + z] is the best dominant.

Putting p = 1, l = m + 1, [[alpha].sub.1] = 1 and [[alpha].sub.i+1] = [[beta].sub.i] (i = 1, 2, ...m) in Corollary 3.12, we obtain the following result.

Corollary 3.13. Let 0 < [eta] [less than or equal to] 1 and 2[mu] [greater than or equal to] v. If f [member of] A, F as defined in (2.8) and

[(F'(z)).sup.[mu]] [(z/F(z)).sup.v] (2[mu]f'(z)/F'(z) - v f(z)/F(z)) < (2[mu] - v) [square root of 1 + z] + z/[2[square root of 1 + z]]

then

[(F'(z)).sup.[mu]] [(z / F(z)).sup.v] < [square root of 1 + z]

and [square root of 1 + z] is the best dominant.

Putting [mu] = v = 1 in the above Corollary 3.13, we have the following example.

Example 3.3. Let 0 < [eta] [less than or equal to] 1. If f [member of] A, F as defined in (2.8) and

[absolute value of zF'(z) / F(z) (2 f'z/F'(z) - f(z)/F(z))] < [square root of 1.22] [approximately equal to] 1.10,

then F [member of] SL.

Proof. The above result can be proved using the technique adopted in the proof of Example 3.2 and hence it is omitted here.

Next we discuss some applications of Theorem 2.1 when L = 1, the multiplier transformation. In Theorem 2.1, the subordination part yields the following corollary by taking [psi](z) = (1 + (1 + 2[alpha])z)/(1 - z), 0 [less than or equal to] [alpha] < 1 and

Re (1 + z[psi]" (z) / [psi]'(z)) > max{0, Re[(v - [mu])([lambda] + p)]}

in place of "[psi] is convex and Re[([mu] - v)([lambda] + p)] [greater than or equal to] 0".

Corollary 3.14. Let 0 [less than or equal to] [alpha] < 1, [lambda] [not equal to] -p be a complex number and Re[([mu] - v)([lambda] + p)] [greater than or equal to] 0. If f [member of] [A.sub.p] and

[[OMEGA].sup.r.sub.I,u,v] (f(z)) ([mu][[OMEGA].sup.r+1.sub.I,1,1](f(z)) - v[[OMEGA].sup.r.sub.I,1,1] (f(z))) < (([mu] - v) [1 + (1 - 2[alpha])z / 1 - z] + [1 / [lambda] + p 2(1 - [alpha])z / [(1 - z).sup.2]],

then

[[OMEGA].sup.r.sub.I,u,v] (f(z)) < [1 + (1 - 2[alpha])z / 1 - z]

and (1 + (1 - 2[alpha])z)/(1 - z) is the best dominant.

Note that for p = 1, [lambda] = 0 and r = 0, we have [I.sub.1] (0, 0)f (z) = f (z), [I.sub.1] (1, 0) f (z) = z f'(z), [I.sub.1] (2, 0)f (z) = z(z f''(z) + f'(z)). Putting these values in Corollary 3.14, we have the following result.

Corollary 3.15. Let 0 [less than or equal to] [alpha] < 1 and [mu] [greater than or equal to] v. If f [member of] A and satisfies

Re [[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](1 + zf"(z)/f'(z)) - vzf'(z)/f(z))] > 2([mu] - v)[alpha] - (1 - [alpha]) / 2,

then

Re [[(f'(z)).sup.[mu]][(z/f(z)).sup.v]] > [alpha].

Proof. The proof is similar to that of the Corollary 3.3 and hence omitted here.

Setting [mu] = v = 1 in Corollary 3.15, we have the following result:

Example 3.4. Let 0 [less than or equal to] [alpha] < 1. If f [member of] A satisfies the differential subordination

Re [zf'(z) / f(z) (1 - zf'(z)/f(z) + zf"(z)/f'(z))] > [[alpha] - 1] / 2,

then f [member of] S* ([alpha]).

Remark 3.2. For [alpha] = 0, the above asserted Example 3.4 reduces to a result obtained by Owa and Obradovic [27, Corollary 2].

Putting [mu] = 1 and v = 0 in Corollary 3.15, we have the following result:

Example 3.5. Let 0 [less than or equal to] [alpha] < 1. If f [member of] A and satisfies

Re[f'(z) + zf''(z)] > [3[alpha] -1] / 2,

then Re f'(z) > [alpha].

Remark 3.3. The above Example 3.5 extends the result [8, Theorem 5] due to Chichra. Further corollary 3.15 reduces to [24, Theorem 2] when [mu] = 0, v = -1 and [alpha] = 1 /3.

If we take [psi](z) = [((1 + z)/(1 - z)).sup.[eta]] with 0 < [eta] [less than or equal to] 1, for the case L = 1, then clearly [psi](z) is convex in the open unit disc D and we have the following corollary from the subordination part of Theorem 2.1.

Corollary 3.16. Let 0 < [eta] [less than or equal to] 1, [lambda] [not equal to] -p be a complex number and Re[([mu] - v)([lambda] + p)] [greater than or equal to] 0. If f [member of] [A.sub.p], and satisfies the subordination

[[OMEGA].sup.r.sub.I,u,v] (f(z)) ([mu][[OMEGA].sup.r+1.sub.I,1,1](f(z)) - v[[OMEGA].sup.r.sub.I,1,1] (f(z)))

< (([mu] - v) + 2[eta]z/([lambda] + p)(1 - [z.sup.2])) [(1 + z / 1 - z).sup.[eta]],

then

[[OMEGA].sup.r.sub.I,u,v] (f(z)) < [(1 + z / 1 - z).sup.[eta]]

and [([1+z / 1-z]).sup.[eta]] is the best dominant.

Putting p = 1, [lambda] = 0 and r = 0 in Corollary 3.16, we obtain the following corollary.

Corollary 3.17. Let 0 < [eta] [less than or equal to] 1 and [mu] [greater than or equal to] v. If f [member of] A and satisfies

[absolute value of arg {[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](1 + zf"(z)/f'(z)) - vzf'(z) / f(z))}] < [delta][pi] / 2,

where

[delta] = [eta] + 1 - 2/[pi] arctan [[mu] - v] / [eta],

then

[absolute value of arg {(f'[(z)).sup.[mu]] [(z/f(z)).sup.v]}] < [eta][pi]/2.

Proof. The proof is much akin to the proof of Corollary 3.5 hence it is left here.

Taking [psi](z) = [square root of 1 + z], convex function in the open unit disc D, as dominant in the subordination part of the Theorem 2.1, we obtain the following corollary.

Corollary 3.18. Let [lambda] [not equal to] -p be a complex number and Re[([mu] - v)([lambda] + p)] [greater than or equal to] 0.

If f [member of] [A.sub.p], and satisfies the subordination

[[OMEGA].sup.r.sub.I,[mu],v] (f(z)) ([mu][[OMEGA].sup.r+1.sub.I,1,1] (f(z)) - v[[OMEGA].sup.r+1.sub.I,1,1] (f(z))) < ([mu] - v) [square root of 1 + z] + z / [2([lambda] + p) [square root of 1 + z]

then

[[OMEGA].sup.r.sub.I,[mu],v] (f(z)) < [square root of 1 + z]

and [square root of 1 + z] is the best dominant.

Putting p = 1, [lambda] = 0 and r = 0 in Corollary 3.18, we have the following corollary.

Corollary 3.19. Let [mu] [greater than or equal to] v. If f [member of] A and satisfies the subordination

[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] ([mu](1 + zf" (z) / f'(z)) - vzf'(z) / f(z)) < ([mu] - v) [square root of 1 + z] + z/[2[square root of 1 + z]],

then

[(f'(z)).sup.[mu]] [(z/f(z)).sup.v] < [square root of 1 + z]

and [square root of 1 + z] is the best dominant.

Example 3.6. If f [member of] A and satisfies

[absolute value of zf' (z) / f(z) (1 + zf'(z)/f'(z) - zf"(z)/f(z))] < 1 /2 [square root of 2] [approximately equal to] 0.35,

then f [member of] SL.

Proof. Putting [mu] = v = 1 in Corollary 3.19 and using the technique used in the proof of Example 3.2, we get the required result.

Received June 13, 2013, Accepted July 22, 2013.

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Department of Applied Mathematics, Delhi Technological University

(Formerly Delhi College of Engineering), Bawana Road, Delhi--110042, India

Virendra Kumar ([double dagger])

Department of Applied Mathematics, Delhi Technological University

(Formerly Delhi College of Engineering), Bawana Road, Delhi--110042, India

V. Ravichandran ([section])

Department of Mathematics, University of Delhi, Delhi--110007, India

* 2010 Mathematics Subject Classification. Primary 30C80, Secondary 30C45.

([dagger]) Corresponding author. E-mail: spkumar@dce.ac.in

([double dagger]) E-mail: vktmaths@yahoo.in

([section]) E-mail: vravi@maths.du.ac.in