# Third-order differential subordination and superordination results for meromorphically multivalent functions associated with the Liu-Srivastava operator.

1. Introduction, Definitions, and Preliminaries

Let H(U) be the class of functions which are analytic in the open unit disk:

U = {z: z [member of] C, [absolute value of] z] < 1}. (1)

For n [member of] N := {1,2,3, ...} and a [member of] C, let

H [a,n] = {/: f [member of] H (U),

f(z) = a + [a.sub.n][z.sup.n] + [a.sub.n+1] [z.sup.n+1] + ...} (2)

and suppose that H = H[1, 1].

Let f and F be members of the analytic function class H(U). The function f is said to be subordinate to F, or F is superordinate to f, if there exists a Schwarz function W(z), analytic in U with

w (0) = 0, [absolute value of m (z)] < 1 (z [member of] U), (3)

such that

f(z) = F(w (z)). (4)

In such a case, we write

f < F or f (z) < F (z). (5)

Furthermore, if the function F is univalent in U, then we have the following equivalence (see, for details, ):

f (z) < F(z) (z [member of] U) [??] f(0) = F(0),

f(U) [subset] F(U). (6)

Let [[SIGMA].sub.p] denote the class of functions of the form

f(z) = 1/2[z.sup.p] + [[infinity].summation over k=1-p] [a.sub.k][z.sup.k] (p [p [member of] N) (7)

which are analytic and multivalent in the punctured unit disk:

[U.sup.*] = {z [member of] C : 0 < [absolute value of z] < 1} = U \ {0}. (8)

For the function f given by (7) and the function g given by

g(z) = 1/2[z.sup.p] + [[infinity].summation over k=1-p] [b.sub.k][z.sup.k] (p [member of] N; z [member of] [U.sup.*]), (9) the Hadamard product (or convolution) f * g of the functions f and g is defined by

(f * g) (z) := 1/2[z.sup.p] + [[infinity].summation over k=1-p] [a.sub.k][b.sub.k][z.sup.k] =: (g * f)(z). (10)

For parameters [[alpha].sub.i] [member of] C (i = 1, 2, ..., q) and [[beta].sub.j] [member of] C \ [Z.sup.-.sub.0] ([Z.sup.-.sub.0] = 0, -1, -2, ...; j = 1, 2, ..., s), the generalized hypergeometric function [sub.q][F.sub.s]([[alpha].sub.1], ..., [[alpoha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]; z) is defined by (see, for example, [2, 3])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

where [(v).sub.k] denotes the Pochhammer symbol defined, in terms of Gamma function, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Recently, Tang et al.  introduced a function

[h.sup.[lambda],[mu].sub.p] ([[alpha].sub.1], ..., [[alpha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]; z) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

In particular, when [lambda] = [mu] = 0, we obtain

[h.sup.0,0.sub.p] ([[alpha].sub.1], ..., [[alpha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]; z) = [h.sub.p] ([[alpha].sub.1], ..., [[alpha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]; z), (14) which was introduced and studied by Liu and Srivastava .

Corresponding to the function

[h.sup.[lambda],[mu].sub.p] ([[alpha].sub.1], ..., [[alpha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]; z) given by (13), we consider a convolution operator

[H.sup.[lambda],[mu].sub.p] ([[alpha].sub.1], ..., [[alpha].sub.q]; [[beta].sub.1], ..., [[beta].sub.s]): [[summation.sub.p] [right arrow] [[summation.sub.p] (15)

defined by the following Hadamard product (or convolution):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16) For the sake of convenience, we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

It is easily verified from definition (16) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

We note that, for [lambda] = [mu] = 0, the operator [H.sup.0,0.sub.p,q,s]([[alpha].sub.1]) reduces to the Liu-Srivastava operator [H.sub.p,q,s]([[alpha].sub.1]) (see [5, 6]; see also ), while the Liu-Srivastava operator is the meromorphic analogous of the Dziok-Srivastava operator (see [8-10]; see also [11, 12]), which includes (as its special cases) the meromorphic analogous of the Carlson-Shaffer convolution operator [L.sub.p](a,c) = [H.sup.0,0.sub.p,2,1] (1, a; c) (see [13, 14]), the meromorphic analogous of the Ruscheweyh derivative operator [D.sup.n.sub.+1] = [L.sub.p](n + p, 1) (see ), and the operator

[J.sub.[delta],p] = [delta]/[z.sup.[delta]+p] [[integral].sup.z.sub.0] [t.sup.[delta]+p-1] f (t) dt = [L.sub.p] ([delta], [delta] + 1) ([delta] > 0) (20)

studied by Uralegaddi and Somanatha .

Let [OMEGA] be any set in C. Also let p be analytic in U and suppose that [psi] : [C.sup.4] x U [right arrow] C. Recently, Antonino and Miller  have extended the theory of second-order differential subordinations in U introduced by Miller and Mocanu  to the third-order case. They determined properties of functions p(z) that satisfy the following third-order differential subordination:

{[psi] (p (z), zp' (z), [z.sup.2] p" (z), [z.sup.3] p'" (z); z): z [member of] U} [subset] [OEMGA]. (21)

We will now recall some definitions and a theorem due to Antonino and Miller , which are required in our next investigations.

Definition 1 (see , p. 440, Definition 1). Let [psi] : [C.sup.4] x U [right arrow] C and h(z) be univalent in U. If p(z) is analytic in U and satisfies the following third-order differential subordination:

[psi] (p (z), zp (z), [z.sup.2]p" (z), [z.sup.3]p'" (z); z) < h (z), (22)

then p(z) is called a solution of the differential subordination. A univalent function q(z) is called a dominant of the solutions of the differential subordination or, more simply, a dominant if p(z) < q(z) for all p(z) satisfying (22). A dominant [??](z) that satisfies [??](z) < q(z) for all dominants q(z) of (22) is said to be the best dominant.

Definition 2 (see , p. 441, Definition 2). Let Q denote the set of functions q that are analytic and univalent on the set [bar.U] E(q), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (23)

is such that

min [absolute value of q' ([xi])] = [rho] > 0 (24)

for [xi] [member of] [partial derivative]U \ E(q). Further, let the subclass of Q for which q(0) = a be denoted by Q(a) and

Q (1) = [Q.sub.1]. (25)

Definition 3 (see , p. 449, Definition 3). Let [OMEGA] be a set in C, q [member of] Q, and n [member of] N \ {1}. The class of admissible functions [[psi].sub.n][[OMEGA], q] consists of those functions [psi] : [C.sup.4] x U [right arrow] C that satisfy the following admissibility condition:

[psi] (r, s, t, u; z) [not member of] [OMEGA] (26)

whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where z [member of] U, [xi] [member of] [partial derivative]U E(q), and k {??]n.

Theorem 4 (see , p. 449, Theorem 1). Let p [member of] H[a,n] with n [member of] N \ {1}. Also let q [member of] Q(a) and satisfy the following conditions:

R([xi)q'' ([xi])/q'([xi]) [??] 0, [absolute value of zp' (z)/q' ([xi])] [??] k, (28) where z [member of] U, [xi] [member of] [partial derivative]U \ E(q), and k [??] n. If [OMEGA] is a set in C, [psi] [member of] [[psi].sub.n] [[OMEGA}, q] and

[psi] (p (z), zp' (z), [z.sup.2] p" (z), [z.sup.3]p'" (z); z) [member of] [OMEGA], (29)

then

p (z) < q (z). (30)

In this paper, following the theory of second-order differential superordinations in the unit disk introduced by Miller and Mocanu , we consider the dual problem of

determining properties of functions p(z) that satisfy the following third-order differential superordination:

[OMEGA] C {[psi](p (z), zp (z), [z.sup.2]p" (z), [z.sup.3]p'" (z);z) : z [member of] U}. (31)

In other words, we determine the conditions on [OMETA], [DELTA], and [psi] for which the following implication holds true:

[OMEGA] C {[psi](p (z), zp' (z), [z.sup.2]p'' (z), [z.sup.3]p"' (z); z) : z [member of] U}

[right arrow] [DELTA] [subset] p(U), (32)

where [DELTA] is any set in C.

If either [OMEGA] or [DELTA] is a simply connected domain, then

(32) can be rephrased in terms of superordination. If p(z) is univalent in U, and if {DELTA] is a simply connected domain with [DELTA] [not equal to] C, then there is a conformal mapping q of U onto [DELTA] such that q(0) = p(0). In this case, (32) can be rewritten as follows:

[OMEGA] C {[psi](p (z), zp' (z), [z.sup.2]p' (z), [zl.sup.3]p"' (z);z) : z [member of] U}

[??] q (z) < p (z). (33)

If [OMEGA] is also a simply connected domain with [OMEGA] = C, then there is a conformal mapping h of U onto [OMEGA] such that h(0) = [psi](p(0), 0, 0,0; 0). In addition, if the function

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z) (34)

is univalent in U, then 33) can be rewritten as

h(z) < [psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z) [??] q (z) < p (z). (35)

There are three key ingredients in the implication relationship

(33): the differential operator [[psi], the set [OMEGA], and the "dominating" function q. If two of these entities were given, one would hope to find conditions on the third entity so that (33) would be satisfied. In this paper, we start with a given set [OMEGA] and a given function q, and we then determine a set of "admissible" operators [psi] so that (33) holds true.

We first introduce the following definition.

Definition 5. Let [psi] : [C.sup.4] x U [right arrow] C and the function h(z) be analytic in U. If the functions p(z) and

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z) (36)

are univalent in U and satisfy the following third-order differential superordination:

h(z) < [psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z), (37)

then p(z) is called a solution of the differential superordination. An analytic function q(z) is called a subordinant of

the solutions of the differential superordination or more simply a subordinant if q(z) < p(z) for p(z) satisfying (37). A univalent subordinant [??](z) that satisfies the condition

q (z) < [??](z) (38)

for all subordinants q(z) of (37) is said to be the best subordinant. We note that the best subordinant is unique up to a rotation of U.

For [OMEGA]a set in C, with [psi] and p as given in Definition 5, we suppose that (37) is replaced by

[OMEGA]C {[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3p]"' (z);z) : z [member of] U}. (39)

Although this more general situation is a "differential containment," yet we also refer to it as a differential superordination, and the definitions of solution, subordinant, and best subordinant as given above can be extended to this more general case.

We will use the following lemma [, p. 445, Lemma D] from the theory of third-order differential subordinations in U to determine subordinants of the third-order differential superordinations.

Lemma 6 (see ). Let p [member of] Q(a), and let q(z) = a + [a.sub.n][z.sup.n] + ... be analytic in U with q(z) [not equal to] a and n [member o f] N \ {1}. If q is not subordinate to p, then there exists points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(i) q([z.sub.0]) = p([[xi].sub.0]),

(ii) R ([[xi].sub.0]p" *[[xi].sub.0])/p' [[xi].sub.0])) [??] 0 and [absolute value of zq'(z)/p'([[xi].sub.0])] [??] m,

(iii) z([z.sub.0]) q'([z.sub.0]) = m), [[xi].sub.0] p' ([[xi].sub.0]),

(iv) R(1 + [z.sub.0] q"([z.sub.0])/q'([z.sub.0])) [??] mR(1 + [[xi].sub.0] p" ([[xi].sub.0])/p' [[xi].sub.0])),

(v) R([z.sup.2.sub.0] q'"([z.sub.0])/q'([z.sub.0])) [??] [m.sup.2]R [[xi].sup.2.sub.0] p"' ([[xi].sub.0])/p'([[xi].sub.0])). 2. Admissible Functions and a Fundamental Result

We next define the class of admissible functions referred to in the preceding section.

Definition 7. Let [OMEGA] be a set in C, q [member of] H[a, n] and q' (z) [not equal to] 0. The class of admissible functions [[psi]'.sub.n][[OMEGA], q] consists of those functions [psi] : [C.sup.4] x [bar.U] [right arrow] C that satisfy the following admissibility condition:

[psi] (r, s, t, u; [xi]) [member of] Q (40)

whenever

r = q (z), s = zq'(z)/m,

R (t/s + 1) [??] 1/m R (zq" (z)/q' (z) +1). (41)

R (u/s) [??] 1/[m.sup.2] R ([z.sup.2] q"' (z)/q' (z)), (42) where z [member of] U, [xi] [member of] [partial derivative]U, and [??] n [??] 2.

If [psi]: [C.sup.2] x [bar.U] [right arrow] C and q [member of]] H[a,n], then the admissibility condition (41) reduces to the following form:

[psi] (q (z),zq' (z)/m; [xi]) [member of] [OMEGA] (z [member of] U; [xi] [member or] [partial derivative]U; m [??] n [>>] 2). (43)

If [psi] : [C.sup.3] x [bar.U] [right arrow] C and q [member of] H[a, n] with q'(z) [not equal to] 0, then the admissibility condition (41) reduces to the following form:

[psi] (r, s, t; [xi]) [member of] [OMEGA] (44)

whenever r = q(z), s = zq'(z)/m, and

R (t/s + 1) [??] 1/m R (zq" (z)/q' (z) + 1) (z [member of U; [xi] [member of] [partial derivative]U; m [??] n [??] 2). (45)

The next theorem is a foundation result in the theory of the third-order differential superordinations in U.

Theorem 8. Let q [member of] H[a, n] and [psi] [member of] [[psi]'.sub.n][[OMEGA], q]. If

[psi](p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z) (46)

is univalent in U and p [member of] Q(a) satisfy the following conditions:

R (zq" (z)/q' (z)) [??] 0, [absolute value of zp' (z)/q' (z)] [??] m (z) [member of] U; [xi] [member of] [partial derivative]U; m [??] n [??] 2), (47)

then

[OMEGA] C {[psi](p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z) : z [member of] U} (48)

implies that

q (z) < p (z). (49)

Proof. Suppose that

q (z) [??] p (z). (50)

Then, by the above lemma, there exists points [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 2 that satisfy conditions (i)-(v) of the above lemma. Using these conditions with r = p([[xi.sub.0]) s = [[xi.sub.0]p'' ([[xi.sub.0]), t = [[xi.sup.2sub.0] p" ([[xi.sub.0]), u = [[xi.sup.3.sub.0] p"' ([[xi.sub.0]) and [xi], = [[xi.sub.0] in Definition 7, we obtain

[psi](p ([[xi.sub.0]), [[xi.sub.0]p' ([[xi.sub.0]), [[xi.sup.2.sub.0] p" ([[xi.sub.0]), [[xi.sup.3.sub.0] p"' ([[xi.sub.0]); [[xi.sub.0]) [member of [OMEGA], (51)

which contradicts (48), so we have

q (z) < p (z). (52)

In the special case when [OMEGA] = C is a simply connected domain and h is a conformal mapping of U onto Q, we denote this class [[psi]'.sub.n](U), q] by [[psi]'.sub.n] [h, q]. The following result is an immediate consequence of Theorem 8.

Theorem 9. Let q [member of] H[a,n]. Also let the function h be analytic in U and suppose that [psi] [member of] [[psi]'.sub.n] [h, q]. If p [member of] Q(a) satisfies condition (47) and

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p'" (z); z) (53)

is univalent in U, then

h(z) < [psi] (p (z), zp' (z), [z.sup.2]p' (z), [z.sup.3]p"' (z); z) (54) implies that

q (z)< p (z). (55)

Theorems 8 and 9 can onlybe used to obtain subordinants of the third-order differential superordination of the forms (48) or 54).

Theorem 10. Let the function h be analytic in U and let [psi] : [C.sup.4] x [bar.U] [right arrow] C. Suppose that the differential equation

[psi] (q (z), zq' (z), [z.sup.2]q" (z), [q.sup.3]p'" (z); z) = h (z) (56)

has a solution q [member of] Q(a). If [psi] [member of] [[psi]'.sub.n] [h, q], p [member of] Q(a), and

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p'" (z); z) (57)

is univalent in U, then (54) implies that

q(z) < p (z) (58)

and q(z) is the best subordinant.

Proof. Since [psi] [member of] [[psi]'.sub.n] [h, q], by applying Theorem 9, we deduce that q is a subordinant of (54). Since q satisfies (56), it is also a solution of the differential superordination (54). Therefore, all subordinants of (54) will be subordinate to q. It follows that q(z) will be the best subordinant of (54).

In the next two sections, by making use of the thirdorder differential subordination results of Antonino and Miller  in the unit disk U and the third-order differential superordination results in U obtained in Section 2 (see, for details, Theorems 8, 9, and 10), we determine certain appropriate classes of admissible functions and investigate some third-order differential subordination and differential superordination properties of meromorphically multivalent functions associated with the operator [H.sup.[lambda],[mu].sub.p,q,s]([[beta].sub.i]) defined by (16). It should be remarked in passing that, in recent years, several authors obtained many interesting results involving various linear and nonlinear convolution operators associated with (second-order) differential subordination and superordination, and the interested reader may refer to several earlier works including (for example)  to 20-23].

3. Third-Order Differential Subordination of the Operator [H.sup.[lambda],[mu].sub.p,q,s]([[beta].sub.i])

We first define the following class of admissible functions, which are required in proving the differential subordination theorem involving the operator [H.sup.[lambda],[mu].sub.p,q,s]([[beta].sub.i]) defined by (16).

Definition 11. Let [OMEGA] be a set in C and q [member of] [Q.sub.i] [union] H. The class of admissible functions [[PHI].sub.H][[OMEGA], q] consists of those functions [phi] : [C.sup.4] x U [right arrow] C that satisfy the following admissibility condition:

[phi] (a, b, c, d; z) [not member of] [OMEGA] (59)

whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (60)

where z [member of] U, [[beta].sub.1] [member of] C \ {0,-l, -2, ...,}, [xi] [member of] [partial derivative]U \ E(q), and k [member of] N \{1}.

Theorem 12. Let [phi] [member of] [[PHI].sub.H][[OMEGA], q]. If the functions f [member of] [[summation.sub.p] and q [member of] [Q.sub.1] satisfy the following conditions:

R ([[xi].sub.q]" ([xi])/q'([xi]) [greater than or equal to] 0, [absolute value of [z.sup.p] [H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z)/q' [xi]] [??] k, (61)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (62)

then

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) + 1) f (z) < q (z). (63

Proof. Define the analytic function p(z) in U by

p (z) = [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z). (64)

Then, differentiating (64) with respect to z and using (18), we have

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) = zp' (z) + [[beta].sub.1] p (z)/[[beta].sub.1]. (65)

Further computations show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (66)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (67)

We now define the transformation from [C.sup.4] to C by

a (r, s, t, u) = r, b (r, s, t, u) = s + [[beta].sub.1]r/[[beta].sub.1],

c (r, s, t, u) = t + 2 ([[beta].sub.1] + 1) s + [[beta].sub.1] ([[beta].sub.1] + 1) r/[[beta].sub.1] ([[beta].sub.1] +

d(r, s, t, u)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (69)

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (70)

The proof will make use of Theorem 4. Using (54) to (57), we find from (70) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (71)

Hence, clearly, (62) becomes

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p'" (z); z) [member of] [OMEGA]. (72)

We note that

t/s + 1 ([[beta].sub.1] + 1) (c - a)/b - a -(2[[beta].sub.1] + 1),

u/s ([[beta].sub.1] + 1) ([[beta].sub.1] + 2) (d - 3c + 3b - a)/b - a. (73)

Thus, the admissibility condition for [psi] [member of] [[PHI].sub.H][[OMEGA], q] in Definition 11 is equivalent to the admissibility condition for [psi] [member of] [[psi].sub.2][[OMEGA], q] as given in Definition 3 with n = 2. Therefore, by using (61) and Theorem 4, we have

p (z) < q (z) (74)

or, equivalently,

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < q (z), (75)

which evidently completes the proof of Theorem 12.

Our next result is an extension of Theorem 12 to the case where the behavior of q(z) on [partial derivative]U is not known.

Corollary 13. Let Q c C and let the function q be univalent in U with q(0) = 1. Suppose also that [phi] [member of] [[PHI].sub.H][[OMEGA], [q.sub.p]] for some [rho] [member of] (0,1), where [q.sub.p](z) = q([rho]z). If the functions f [member of] [[summation].sub.p] and [q.sub.[rho]] satisfy the following conditions:

R([[xi].sup.q".sub.[rho]] [xi]/[q'.sub.[rho]] ([xi])) [??] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (76)

then

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < q (z). (77)

Proof. We note from Theorem 12 that

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < [q.sub.[rho] (z). (78)

The result asserted by Corollary 13 is now deduced from the following subordination property:

[q.sub.[rho]] (z) < q (z). (79)

If [OMEGA} = C is a simply connected domain, then [OMEGA] = h( U) for some conformal mapping h(z) of U onto [OMEGA]. In this case, the class [[PHI].sub.H][h(U),q] is written as [[PHI].sub.H][h,q]. The following two results are immediate consequences of Theorem 12 and Corollary 13.

Theorem 14. Let [phi] [member of] [[PHI].sub.H][h, q]. If the functions f [member of] [[summation].sub.p] and q [member of] [Q.sub.1] satisfy the following conditions:

R([xi]q" ([xi])/q'([xi])) [??] 0, [absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z)]/q' ([xi])] [??] k, (80)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (81)

then

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < q (z). (82)

Corollary 15. Let [OMEGA] [subset] C and let the function q be univalent in U with q(0) = 1. Suppose also that [phi] [member of] [[PHI].sub.H][h, [q.sub.[rho]] for some [rho] [member of] (0,1), where [q.sub.[rho]](z) = q([rho]z). If the functions f [member of] [[summation].sub.p] and [q.sub.[rho]] satisfy the following conditions:

R([xi][q".sub.[rho]] ([xi])/[q'.sub.[rho]] ([xi])) [??] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (83)

then

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < q (z). (84)

Our next theorem yields the best dominant of the differential subordination (70).

Theorem 16. Let the function h be univalent in U. Also let f : [C.sup.4] x U [right arrow] C and y be given by (70). Suppose that the differential equation

[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p'" (z); z) = h(z) (85)

has a solution q(z) with q(0) = 1, which satisfies condition (61). If the function f [member of] [[summation].sub.p] satisfies condition (81) and the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (86)

is analytic in U, then

[z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < q (z). (87)

and q(z) is the best dominant.

Proof. By applying Theorem 12, we deduce that q is a dominant of (81). Since q satisfies (85), it is also a solution of (81). Therefore, q will be dominated by all dominants. Hence q is the best dominant.

In view of Definition 11, in the particular case when q(z) = 1 + Mz (M > 0), the class [[PHI].sub.H][[OMEGA], q] of admissible functions, denoted simply by [[PHI].sub.H][[OMEGA] M], is described below.

Definition 17. Let [OMEGA] be a set in C, [[beta].sub.1] [member of] C \ {0, -1, -2, ...}, and M > 0. The class [[PHI].sub.H][[OMEGA], M] of admissible functions consists of those functions [phi] : [C.sup.4 x U [right arrow] C such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (88)

whenever z [member of] U, R([Le.sup.-i[theta]]) [??] (k - 1)kM and R([Ne.sup.-i[theta]) [??] 0 for all [theta] [member of] R and k [member of] N \ {1}.

Corollary 18. Let [phi] [member of] [[PHI].sub.H][[OMEGA], M]. If the function f [member of] [[summation].sub.p] satisfies the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (89)

then

[absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) -1] < M (M > 0). (90)

In the special case when

[OMEGA] = q (U) = {[omega]: [absolute value of [omega] -1] < M (M > 0)}, (91)

the class [[PHI].sub.H][[OMEGA], M] is denoted, for brevity, by [[PHI].sub.H][[OMEGA], M]. Corollary 18 can now be rewritten in the following form.

Corollary 19. Let [phi] [member of] [[PHI].sub.H][[OMEGA], M].If thefunction f [member of] [[summation].sub.p] satisfies the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (92)

then

[absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) -1] < M (M > 0). (93)

Corollary 20. Let [[beta].sub.1] [member of] C\{0, -1, -2, ...} with R([[beta].sub.1] [??] -1/2 and M > 0. If the function f [member of] [[summation].sub.p] satisfies the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (94)

then

[absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) -1] < M. (95)

Proof. Corollary 20 follows from Corollary 19 by setting

[phi] (a, b, c, d; z) = b = 1 + k + [[beta.sub.1]/[[beta].sub.1] [Me.sup.i[theta]]. (96)

Corollary 21. Let [beta].sub.1] [member of] C \ {0, -1, -2, ...}, k [member of] N \ {1}, and M > 0. If the function f [member of] [[summation].sub.p] satisfies the following conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (97)

then

[absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) -1] < M. (98)

Proof. Let

[phi] (a, b, c, d; z) = d - c, [OMEGA] = h(U), (99)

where

h(z) = Mz/[absolute value of [[beta].sub.1]] (M > 0). (100)

In order to use Corollary 18, we need to show that [phi] [member of] [[PHI].sub.H][[OMEGA], M]; that is, the admissibility condition (88) is satisfied. This follows easily, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (101)

whenever z [member of] U, R([Le.sup.-i[thera]]) [??](k - 1)kM, and R([Ne.sup.-i[theta]]) [??] 0 for all [theta] [member of] R and k [member of] N \ {1}. The required result now follows from Corollary 18.

4. Third-Order Differential Superordination of the Operator] [H.sup.[lambda],[mu].sub.p,q,s]([[beta].sub.i])

In this section, we obtain the third-order differential superordination results for meromorphically multivalent functions associated with the operator H.sup.[lambda],[mu].sub.p,q,s]([[beta].sub.i]) defined by (16). Because of this, the class of admissible functions is given in the following definition.

Definition 22. Let [OMEGA] be a set in C and q [member of] H with q(z) [not equal to] 0. The class of admissible functions [[PHI].sub.H][[OMEGA], q] consists of those functions [phi] : [C.sup.4] x [bar.U] [right arrow] C that satisfy the following admissibility condition:

[phi] (a, b, c, d; [xi]) [member of] [OMEGA] (102)

whenever

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (103)

where z [member of] U, [[beta].sub.1] [member of] C \ {0, -1, -2, ...}, [xi] [member of] [partial derivatiave]U, and me N \ {1}.

Theorem 23. Let [phi] [member of] [[PHI]'.sub.H][[OMEGA], q]. If the functions f [member of] and [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) [member of] [Q.sub.1] satisfy the following conditions:

R([xi)q'' ([xi])/q'([xi]) [??] 0, [absolute value of [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f z)/q' z] [??] m, (104)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (105)

is univalent

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (106)

implies that

q (z) < [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z). (107)

Proof. Let the function p(ft) be defined by (64) and [psi] by (70). Since [phi] [member of] [[PHI].sub.H][[OMEGA], q], 71) and (106) yield

[OMEGA] [subset] {[psi] (p (z), zp' (z), [z.sup.2]p" (z), [z.sup.3]p"' (z); z): [member of] U}. (108)

We see from (68) and (69) that the admissible condition for [phi] [member of] [[PHI]'.sub.H][[OMEGA], q] in Definition 22 is equivalent to the admissible condition for y as given in Definition 7 with n = 2. Hence [psi] [member of] [[PHI]'.sub.H][[OMEGA] [Q, q], and by using (104) and Theorem 8, we have

q (z) < p (z) (109)

or, equivalently,

q (z) < [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z). (110)

which evidently completes the proof of Theorem 23.

If [OMEGA] = C is a simply connected domain and [OMEGA] = h(U) for some conformal mapping h(ft) of U onto [OMEGA], then the class [[PHI]'.sub.H][h (U), q] is written simply as [[PHI]'.sub.H][h, q]. With proceedings similar as in the preceding section, the following result is an immediate consequence of Theorem 23.

Theorem 24. Let [phi] [member of] [[PHI]'.sub.H][ h, q]. Also let the function h be analytic in U. If the functions f [member of] and [[summation].sub.p] and [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) [member of] [Q.sub.1] satisfy condition (104) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (111)

is univalent in U, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (112)

implies that

q (z) < [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z). (113)

Theorems 23 and 24 can onlybe used to obtain subordinations involving the third-order differential superordination of the forms (106) or 112). The following theorem proves the existence of the best subordinant of (112) for a suitable chosen [phi].

Theorem 25. Let the function h be analytic in U, and let [phi] : [C.sup.4[ x [bar.U] [right arrow] C and y be given by (70). Suppose that the differential equation

[psi] (q (z), zq' (z), [z.sup.2]q" (z), [q.sup.3]p'" (z); z) = h (z) (114)

has a solution q(z) [member of] [Q.sub.1]. If the functions f [member of] [[summation].sub.p] and [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) [member of] [Q.sub.1] (104) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (115)

is univalent in U, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (116)

implies that

q (z) < [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) (117)

and q is the best subordinant.

Proof. The proof of Theorem 25 is similar to that of Theorem 16 and it is being omitted here.

By combining Theorems 14 and 24, we obtain the following sandwich-type result.

Corollary 26. Let the functions [h.sub.1] and [q.sub.1] be analytic functions in U. Also let thefunction h2 be univalent in U, [q.sub.2] [member of [Q.sub.1] with [q.sub.1](0) = [q.sub.2](0) = 1 and [phi] [member of] [[PHI].sub.H][[h.sub.2], [q.sub.2]][intersection] [[PHI]'.sub.H][[h.sub.1], [q.sub.1]]. If the function f [member of] [[summation].sub.p], [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) [member of] {Q.sub.1] H and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (118)

is univalent in U, and the conditions (61) and (104) are satisfied, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (119)

implies that

[q.sub.1] < [z.sup.p][H.sup.[lambda],[mu].sub.p,q,s] ([[beta].sub.1]) f (z) < [q.sub.2] (z). (120)

Remark 27. By setting [lambda] = [mu] = 0 in all results of this paper, we can obtain the corresponding results for the well-known Liu-Srivastava operator [H.sub.p,q.s]([[beta].sub.1]).

5. Concluding Remarks and Observations

In our present investigation, we have derived several third-order differential subordination and superordination results for meromorphically multivalent functions in the punctured unit disk involving the operator [H.sub.p,q.s]([[beta].sub.1]) defined by (16) with respect to the parameter [[beta].sub.1] [member of] C \ {0,-1, -2, ...}, which is associated with the Liu-Srivastava operator [H.sub.p,q,s]([[beta].sub.1]). Our results have been obtained by considering suitable classes of admissible functions. Furthermore, if we use relation (19), we can obtain the corresponding third-order differential subordination and superordination results for the operator [H.sub.[lambda],[mu].sub.p,q,s]([[alpha].sub.1]) with respect to the parameter [[alpha].sub.1] [member of] C and here we choose to omit the details involved.

http://dx.doi.org/10.1155/2014/792175

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher

School Doctoral Foundation of China under Grant 20100003110004, the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117, and the Higher School Foundation of Inner Mongolia of China under Grant NJZY13298. The authors would like to thank Professors Om P. Ahuja and V. Ravichandran for their valuable suggestions and the referees for their careful reading and helpful comments to improve their paper.

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Huo Tang, (1,2) H. M. Srivastava, (3) Shu-Hai Li, (1) and Li-Na Ma (1)

(1) School of Mathematics and Statistics, Chifeng University, Chifeng, InnerMongolia 024000, China

(2) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

(3) Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4

Correspondence should be addressed to Huo Tang; thth2009@tom.com and H. M. Srivastava; harimsri@math.uvic.ca

Received 14 April 2014; Revised 7 June 2014; Accepted 8 June 2014; Published 8 July 2014

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