# SANDWICH RESULTS FOR P-VALENT MEROMORPHIC FUNCTIONS ASSOCIATED WITH HURWITZ-LERECH ZETA FUNCTION.

1. INTRODUCTION

Let denote by H(U) the space of all analytical functions in the unit disk U = {z [member of] C: |z| < 1}, and for a [member of] C, n [member of] N*, we denote

H[a, n] = {f [member of] H(U): f (z) = a + [a[.sub.]n][z[.sup.]n] +...}.

Let denote the class of functions

[A[.sub.]n] = {f [member of] H(U): f (z) = z + [a[.sub.]n+1][z[.sup.]n+1] +...},

and let A [equivalent to] [A[.sub.]1].

If f, F [member of] H(U) and F is univalent in U we say that the function f is subordinate to F, or F is superordinate to f, written f (z) [??] F(z), if f (0) = F(0) and f (U) [??] F(U).

Letting [phi]: [C[.sup.]3] X U [right arrow] C, h [member of] H(U) and q [member of] H[a, n], in  the authors determined conditions on [phi] such that

h(z) [??] [phi](p(z),zp', [z[.sup.]2]p"(z); z) implies q(z) [??] p(z),

for all p functions that satisfy the above superordination. Moreover, they found sufficient conditions so that the q function is the largest function with this property, called the best subordinant of this superordination.

Using the principle of subordination, Miller et al.  investigated some subordination theorems involving certain integral operators for analytic functions in U (see also [2, 18]). Moreover, Miller and Mocanu  considered the differential superordinations as the dual concept of differential subordinations (see also ).

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

(1.1) f(z) = [z[.sup.]-p] + [[infinity].summation over (n=1-p)] [a[.sub.]n][z[.sup.]n] (n,p [member of] N = {1, 2,...}),

which are analytic and p-valent in the punctured unit disc U = {z [member of] C: 0 < |z| < 1} = U \ {0}. We note that [SIGMA] [equivalent to] [[SIGMA][.sub.]1] the class of univalent meromorphic fuctions. For the functions f [member of] [[SIGMA][.sub.]p] given by (1.1) and g [member of] [[SIGMA][.sub.]p] given by

g(z) = [z[.sup.]-p] + [[infinity].summation over (n=1-p)] [b[.sub.]n][z[.sup.]n] (n,p [member of] N),

the Hadamard (or convolution) product of f and g is given by

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

The general Hurwitz-Lerech Zeta function [PHI](z, s, b) is defined by (see )

[PHI](z, s, d) = [[infinity].summation over (n=0)] [z[.sup.]n]/[(n+d)[.sup.]s],

with [mathematical expression not reproducible], s [member of] C when |z| < 1 and Re s > 1 when |z| = 1 (all the powers are principal ones).

Several interesting properties and characteristics of the above defined Hurwitz-Lerech Zeta function may be found in the investigations by several authors (see , , , ).

Now, defining the function [mathematical expression not reproducible] by

[mathematical expression not reproducible]

we could introduce the linear operator

[mathematical expression not reproducible]

defined by

[mathematical expression not reproducible]

We note that

(1.2) [mathematical expression not reproducible]

where all the powers are principal ones, and using this form of the operator [mathematical expression not reproducible] it is easy to verify that

(1.3) [mathematical expression not reproducible]

Also, we note that

(i) [mathematical expression not reproducible];

(ii) [mathematical expression not reproducible]

Moreover, we could easily check that for all f [member of] [[SIGMA][.sub.]p] we have

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

We remark the following special cases of the operator [mathematical expression not reproducible]:

(i) [mathematical expression not reproducible] (see [14, p. 11 and p. 389]);

(ii) [mathematical expression not reproducible]

(see Aqlan et al. );

(iii) [mathematical expression not reproducible]

(see El-Ashwah and Aouf );

(iv) [mathematical expression not reproducible]

(see El-Ashwah ).

In the present paper we obtain some type of subordination and superordination preserving properties for the linear operators [mathematical expression not reproducible] defined by (1.2), and the corresponding sandwich-type theorem.

2. PRELIMINARIES

To prove our main results, we will need the following definitions and lemmas presented in this section.

A function L(z; t): U X [0, +[infinity]) [right arrow] C is called a subordination (or a Loewner) chain if L(x;t) is analytic and univalent in U for all t [greater than or equal to] 0, and L(z; s) [??] L(z; t) when 0 [less than or equal to] s [less than or equal to] t.

The next well-known lemma gives a sufficient condition so that the L(z; t) function will be a subordination chain.

Lemma 2.1. [12, p. 159] Let L(z; t) = [a[.sub.]1](t)z + [a[.sub.]2](t)[z[.sup.]2] +..., with [a[.sub.]1](t) [not equal to] 0 for all t [greater than or equal to] 0 and [mathematical expression not reproducible]. Suppose that L(x; t) is analytic in U for all t [greater than or equal to] 0, L(z; x) is continuously differentiate on [0, +[infinity]) for all z [member of] U. If L(z; t) satisfies

Re [z [partial derivative]L/[partial derivative]z/[partial derivative]L/[partial derivative]t] > 0, z [member of] U, t [greater than or equal to] 0,

and

|L(z; t)| [less than or equal to] [K[.sub.]0] |[a[.sub.]1](t)|, |z| < [r[.sub.]0] < 1, t [greater than or equal to] 0

for some positive constants [K[.sub.]0] and [r[.sub.]0], then L(z; t) is a subordination chain.

We denote by K([alpha]), [alpha] < 1, the class of convex functions of order [alpha] in the unit disk U, i.e.

K([alpha]) = {f [member of] A:Re [1 + zf''(z)/f'(z)] > [alpha], z [member of] U}.

In particular, the class K [equivalent to] K(0) represents the class of convex (and univalent) functions in the unit disk.

Lemma 2.2. , [15, Theorem 2.3i, p. 35] Suppose that the function H: [C[.sup.]2] [right arrow] C satisfies the condition

Re H(is, t) [less than or equal to] 0,

for all s, t [member of] R with t [less than or equal to] -n(1 + [s[.sup.]2])/2, where n is a positive integer. If the function p(z) = 1 + [p[.sub.]n][z[.sup.]n] +...is analytic in U and

Re H(p(z),zp'(z)) > 0, z [member of] U,

then Re p(z) > 0, z [member of] U.

The next result deals with the solutions of the Briot-Bouquet differential equation (2.1), and more general forms of the following lemma may be found in [14, Theorem 1].

Lemma 2.3.  Let [beta], [gamma] [member of] C with [beta] [not equal to] 0 and let h [member of] H(U), with h(0) = c. If Re[[beta]h(z) + [gamma]] > 0, z [member of] U, then the solution of the differential equation

(2.1) q(z) + zq'(z)/[beta]q(z)+[gamma]=h(z)

with q(0) = c, is analytic in U and satisfies Re[[beta]q(z) + [gamma]] > 0, z [member of] U.

As in , let denote by Q the set of functions f that are analytic and injective on U \ E(f), where

[mathematical expression not reproducible]

and such that f'([zeta]) = 0 for [zeta] [member of] [partial derivative]U \ E(f).

Lemma 2.4. [16, Theorem 7] Let q [member of] H[[alpha], 1], let [chi]: [C[.sup.]2] [right arrow] C and set [chi](q(z), zq'(z)) [equivalent to] h(z). If L(z; t) = [chi](q(z), tzq'(z)) is a subordination chain and p [member of] H[[alpha], 1] [intersection] Q, then

h(z) [??] [chi](p(z), zp'(z)) implies q(z) [??] p(z).

Furthermore, if [chi](q(z),zq'(z)) = h(z) has a univalent solution q [member of] Q, then q is the best subordinant.

Like in  and , let [ohm] [subset] C, q [member of] Q and n be a positive integer. Then, the class of admissible functions [[psi][.sub.]n][[ohm],q] is the class of those functions [psi]: [C[.sup.]3] X U [right arrow] C that satisfy the admissibility condition

[psi](r,s,t; z) [??] [ohm]

whenever r = q([zeta]), s = m[zeta]q'([zeta]), Re t/s+ 1 [greater than or equal to] m Re [[zeta]q''([zeta])/q'([zeta])+1], z [member of] U, [zeta] [member of] [partial derivative]U \ E(q)

and m [greater than or equal to] n. This class will be denoted by [[psi][.sub.]n][[ohm], q].

We write [psi][[ohm],q] [equivalent to] [[psi][.sub.]1][[ohm],q]. For the special case when [ohm] [??] C is a simply connected domain and h is a conformal mapping of U onto [ohm], we use the notation [[psi][.sub.]n][h, q] [equivalent to] [[psi][.sub.]n][[ohm], q].

Remark 2.1. If [psi]: [C[.sup.]2] X U [right arrow] C, then the above defined admissibility condition reduces to

[psi](q([zeta]),mC[zeta]'([zeta]); z) [??] [ohm],

when z [member of] U, [zeta] [member of] [partial derivative]U \ E(q) and m [greater than or equal to] n.

Lemma 2.5. ,  Let h be univalent in U and [psi]: [C[.sup.]3] X U [right arrow] C. Suppose that the differential equation

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

has a solution q, with q(0) = a, and one of the following conditions is satisfied:

(i) q [member of] Q and [psi] [member of] [psi][h, q]

(ii) q is univalent in U and [psi] [member of] [psi][h, [q[.sub.][rho]], for some [rho] [member of] (0,1), where [q[.sub.][rho]](z) = q([rho]z) or

(iii) q is univalent in U and there exists [[rho][.sub.]0] [member of] (0,1) such that [psi] [member of] [psi][[h[.sub.][rho]], [q[.sub.][rho]]] for all [rho] [member of] ([[rho][.sub.]0],1), where [h[.sub.][rho]](z) = h([rho]z) and [q[.sub.][rho]](z) = q([rho]z).

If p(z) = a + [a[.sub.]1]z +...[member of] H(U) and [psi](p(z), zp'(z), [z[.sup.]2]p''(z); z) [member of] H(U), then

[psi](p(z), zp'(z), [z[.sup.]2]p''(z); z) [??] h(z) implies p(z) [??] q(z)

and q is the best dominant.

3. MAIN RESULTS

Unless otherwise mentioned, we assume throughout this paper that [mathematical expression not reproducible], with [d[.sub.]1], [d[.sub.]2] [member of] R, s [member of] C and p [member of] N.

We begin by proving the following subordination theorem:

Theorem 3.1. Let [alpha] < 1 and [mathematical expression not reproducible], with Re d > 1 - [alpha]. For a given function g [member of] [[SIGMA][.sub.]p], suppose that

(3.1) Re [1+ z[phi]''(z)/[phi]'(z)] > [delta], x [member of] U,

where

(3.2) [mathematical expression not reproducible]

and

(3.3)

[mathematical expression not reproducible]

If f [member of] [[SIGMA][.sub.]p] such that

[mathematical expression not reproducible]

then

[mathematical expression not reproducible]

and the function [mathematical expression not reproducible] is the best dominant.

Proof. If we denote

[mathematical expression not reproducible]

and

(3.4) [mathematical expression not reproducible]

then we need to prove that [phi](z) [??] [phi](z) implies F(z) [??] G(z).

Differentiating the second part of the relation (3.4), by using the identity (1.3) we have

[mathematical expression not reproducible]

and replacing the left-hand side of the above relation in (3.2) we get

(3.5) d[phi](z) = ([alpha] - 1 + d)G(z) + (1 - [alpha])zG'(z).

If we let q(z) = 1 +zG''(z)/G''(z), by differentiating (3.5) we have

dz[phi]'(z) = (1 - [alpha])zG'(z) [q(z) +[alpha]-1+d/1-[alpha]]

and by computing the logarithmical derivative of the above equality we deduce

(3.6) q(z) + zq'(z)/q(z) + [alpha]-1+d/1-[alpha] = 1 + z[phi]''(z)/[phi]'(z) [equivalent to] h(z). q(z) + [alpha]-[phi]'(z)

From (3.1), using the assumptions [alpha] < 1 and [d[.sub.]1] = Re d > 1 - [alpha], we have

Re [h(z) + [alpha] - 1 + d/1 - [alpha]] > - [delta] + [alpha]- 1[greater than or equal to] 0, z [member of] U,

and by using Lemma 2.3 we conclude that the differential equation (3.6) has a solution q [member of] H(U), with q(0) = h(0) = 1.

Now we will use Lemma 2.2 to prove that, under our assumption, the inequality

(3.7) Re q(z) > 0, z [member of] U,

holds. Let us put

(3.8) H (u, v) = u +u/u+ [alpha]-1+d/1-[alpha] + [delta],

where [delta] is given by (3.3). From the assumption (3.1), according to (3.6), we obtain

(3.9) Re H(q(z), zq(z)) > 0, z [member of] U,

and we proceed to show that Re H(is, t) [less than or equal to] 0 for all s,t [member of] R, with t [less than or equal to] -(1 + [s[.sup.]2])/2.

From (3.8), using the assumptions [alpha] < 1 and b1 = Re b > -[alpha], we have

[mathematical expression not reproducible]

where

E(s)=([alpha] - 1 + [d[.sub.]1]/1-[alpha] -2[delta])[s[.sup.]2]-4[d[.sub.]2][delta]/1-[alpha] - 2[delta] [|[alpha] - 1 + d|[.sup.]2]/[(1-[alpha])[.sup.]2] [alpha] - 1 + [d[.sub.]1]/1-[alpha],

and [d[.sub.]2] = Im d. It is well-known that the second order polinomial function E(s) is non-negative for all s [member of] R, if and only if

(3.10) [DELTA] [less than or equal to] 0 and [alpha]-1 + [d[.sub.]1]/1-[alpha] - 2[delta] > 0,

where [DELTA] is the discriminant of E(s), i.e.

[DELTA] = 4([alpha] - 1 + [d[.sub.]1])/[(1-[alpha])[.sup.]2] {4([alpha]-1+[d[.sub.]1])[[delta][.sup.]2][[(1 - [alpha])[.sup.]2] + [|[alpha] - 1 + d|[.sup.]2]]/1-[alpha] [delta] + [alpha] +[d[.sub.]1]}

We may easily check that the value of [delta] given by (3.3) is the greater one for which [DELTA] [less than or equal to] 0. Since this value of [delta] satisfies the second part of the conditions (3.10), it follows that Re H(is,t) [less than or equal to] 0 for all s,t [member of] R, with t [less than or equal to] -(1 + [s[.sup.]2])/2.

Form (3.9), according to Lemma 2.2, we deduce that the inequality (3.7) holds, hence G [member of] K, that is G is a convex (and univalent) function in the unit disk, hence the following well-known growth and distortion sharp inequalities (see ) are true:

r/1+r[less than or equal to]|G(z)|[less than or equal to] r/1-r if |z|[less than or equal to] r,

1/[(1+r)[.sup.]2] [less than or equal to]|G'(z)i[less than or equal to] 1/[(1-r)[.sup.]2], if |z|[less than or equal to]r.

If we let

(3.11) L(z; t) =: [alpha]-1+d/d G(z) + (1-[alpha])(1+t)/d zG''(z),

from (3.5) we have L(z; 0) = [phi](z). Denoting L(z; t) = [a[.sub.]1](t)z +..., then

[a[.sub.]1](t)[partial derivative]L(0; t)/[partial derivative]z [alpha] - 1 + d + (1 - [alpha])(1 + t)/d G'(0) =[alpha] - 1 + d + (1 - [alpha])(1 + t)/d

hence [mathematical expression not reproducible], and because [alpha] < 1 and Re d > 1 - [alpha] we obtain [a[.sub.]1](t) [not equal to] 0, [for all]t [greater than or equal to] 0.

From (3.11) we may easily deduce the equality

Re [z [partial derivative]L/[partial derivative]z/[partial derivative]L/[partial derivative]t] = Re [[alpha]- +d/1-[alpha] +(1+t)(1+ zG''(z)/G'(z))]=[alpha]-1 +Re d/1-[alpha] + (1+t) Re q(z).

Using the inequality (3.7) together with the assumptions [alpha] < 1 and Re d > 1 - [alpha], the above relation yields that

Re [z[partial derivative]L/[partial derivative]z/z[partial derivative]L/[partial derivative]t] > 0, [for all]z [member of] U, [for all] t [greater than or equal to] 0.

From the definition (3.11), for all t [greater than or equal to] 0 we have

(3.12) |L(z;t)|/|[a[.sub.]1](t)|[less than or equal to] |[alpha] - 1 + d| |G(z)| + |1 - [alpha]||1 + t| |zG'(z)|/d +(1 - [alpha])t|.

Using the right-hand sides of these inequalities in (3 12), we deduce that

|L(z; t)|/|[a[.sub.]1](t)|[less than or equal to]|[alpha] - 1 + d|/|1-[alpha]| r/1-r [[phi][.sub.]r](t)+r/[(1-r)[.sup.]2], |z|[less than or equal to] r, [for all]t[greater than or equal to]0,

(3.13) [mathematical expression not reproducible]

where

[mathematical expression not reproducible]

Since Re d/1-[alpha] 0 whenever Re d > 1 - [alpha] and [alpha] < 1, it follows

|t + d/1-[alpha]| [greater than or equal to] |d/1-[alpha]|, [for all]t [greater than or equal to] 0,

hence

(3.14) [phi](t) [less than or equal to] |1-[alpha]/d|, t [greater than or equal to] 0.

Moreover, since Re d/1-[alpha]> 1 whenever Re d > 1 - [alpha] and [alpha] < 1, we obtain

|t+1|/|t+d/1-[alpha]| <1 [for all]t [greater than or equal to] 0,

hence

(3.15) [[phi][.sub.]2](t) <1, t [greater than or equal to] 0.

Using the inequalities (3.14) and (3.15), from (3.13) we deduce that

|L(z; t)|/|[[alpha][.sub.]1](t)| < r/[(1 - r)[.sup.]2] + |[alpha]- 1 + d/d] r/1-r,|z| [less than or equal to] r [for all]t [greater than or equal to] 0

hence the second assumption of Lemma 2.1 holds, and according to this lemma we conclude that the function L(z; t) is a subordination chain.

Now, by using Lemma 2.5, we will show that F(z) [??] G(z). Without loss of generality, we can assume that [phi] and G are analytic and univalent in U and G'([zeta]) [not equal to] 0 for |[zeta]| = 1. If not, then we could replace [phi] with [[phi][.sub.][rho]](z) = [phi]([rho]z) and G with [G[.sub.][rho]](z) = G([rho]z), where [rho] [member of] (0,1). These new functions will have the desired properties and we would prove our result using part (iii) of Lemma 2.5.

With our above assumption, we will use part (i) of the Lemma 2.5. If we denote by [psi](G(z), zG'(z)) = [phi](z), we only need to show that [psi] [member of] [psi][[phi], G], i.e. [psi] is an admissible function. Because

[psi](G([zeta]), m[zeta]G'([zeta])) = [alpha]-1+d/d G(z) +(1-[alpha])(1+t)/d zG'(z) = L([zeta]; t),

where m =1 +1, t [greater than or equal to] 0, since L(z; t) is a subordination chain and [phi](z) = L(z; 0), it follows that

[psi](G([zeta]),m[zeta]G'(C)) [??] [phi](U).

According to the Remark 2.1 we have [psi] [member of] [psi][[phi], G], and using Lemma 2.5 we obtain that F(z) [??] G(z) and, moreover, G is the best dominant.

Remark 3.1. It is easy to check that the values of [delta] given by (3.3) satisfies the inequality 0 < [delta] [less than or equal to] 1/2, whenever [alpha] < 1 and Re d > 1 - [alpha].

For the special case d = 1, s = -1 and p =1, taking [beta]:= 1 - [alpha], Theorem 3.1 reduces to:

Corollary 3.1. Let 0 < [beta] < 1 and for a given function g [member of] [SIGMA] suppose that the inequality (3.1) holds, where

(3.16) [phi](z) = [z[.sup.]2] [[beta]zg'(z) + (1 + [beta]) g(z)],

and

(3.17) [mathematical expression not reproducible]

If f [member of] [SIGMA] such that

[z[.sup.]2] [[beta]zf'(z) + (1 + [beta])f (z)] [??] [z[.sup.]2] [[beta]zg'(z) + (1 + [beta])g(z)],

then

[z[.sup.]2]f (z) [??] [z[.sup.]2]g(z),

and the function g is the best dominant.

Now we will prove a dual of Theorem 3.1, in the sense that the subordinations are replaced by superordinations.

Theorem 3.2. Let [alpha] < 1 and [mathematical expression not reproducible], with Re d > 1 - [alpha]. For a given function g [member of] [[SIGMA][.sub.]p], suppose that the function [phi] defined by (3.2) satisfies the condition (3.1), with [delta] given by (3.3).

Let f [member of] [[SIGMA][.sub.]p] such that [mathematical expression not reproducible] is univalent in U and [mathematical expression not reproducible]. Then,

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

and the function [mathematical expression not reproducible] is the best subordinant.

Proof. Denoting

[mathematical expression not reproducible]

and

(3.18) [mathematical expression not reproducible]

then we need to prove that [phi](z) [??] [phi](z) implies G(z) [??] F(z).

If we differentiate the second part of the relation (3.18), using the identity (1.3) we obtain

[mathematical expression not reproducible]

Replacing the left-hand side of the above relation in (3.2) we have

(3.19) [phi](z) = [alpha]-1 + d/d G(z) + 1-[alpha]/d zG'(z).

If we let q(z) = 1+ zG''(z)/G''(z), like in the proof of Theorem 3.1 it follows that the inequality (3.7) holds, i.e. Re q(z) > 0 for all z [member of] U.

Letting

(3.20) L(z; t) = [alpha]-1 + d/d G(z) + (1-[alpha])t/d zG'(z),

from (3.19) we have L(z; 1) = [phi](z). Thus, L(z; t) = [a[.sub.]1](t)z +..., and then

[a[.sub.]1](t)[partial derivative]L(0; t)/[partial derivative]z =[alpha] - 1 + d + (1 - [alpha])t/d G'(0) = ([alpha] - 1 + d + (1 - [alpha])t/d

hence [mathematical expression not reproducible], and because [alpha] < 1 and Re d > 1 - [alpha] we obtain [a[.sub.]1](t) [not equal to] 0, [for all]t [greater than or equal to] 0.

From (3.20), a simple computation shows that

Re [z [partial derivative]L/[partial derivative]z/[partial derivative]L/[partial derivative]t] =Re [[alpha] - 1 + d /1-[alpha] +t (1+ zG''(z)/G''(z)] [alpha] - 1 + Re d/1-[alpha] +t Re q(z).

Since we already mentioned that the inequality (3.7) holds, combining with the assumptions [alpha] < 1 and Re b > -[alpha], the above relation implies that

Re [z[partial derivative]L/[partial derivative]z/[partial derivative]L/[partial derivative]t] > 0, [for all]z [member of] U, [for all]t [greater than or equal to] 0.

Also, for all t [greater than or equal to] 0 we have

(3.21) |L(z;t)|/|[a[.sub.]1] (t)| [less than or equal to] |[alpha] - 1 + d| |G(z)| + |1 - [alpha]||t| |zG''(z)|/|[alpha] - 1 + d + (1 - [alpha])t|.

and from the right-hand sides of these inequalities in (3 12), we obtain that

(3.22) |L(z; t)|/|[a[.sub.]1] (t)|[less than or equal to] |[alpha] - 1 + d|/|1-[alpha] | r/1-r [[phi][.sub.]1](t) +r/[(1-r)[.sup.]2] [[phi][.sub.]2](t),|z|[less than or equal to] r,[for all]t[greater than or equal to] 0,

where

[[phi][.sub.]1](t)= 1/|t+[alpha]-1+d/1-[alpha]| and [[phi][.sub.]2](t) = |t|/|t+[alpha]-1+d/1-[alpha]|

Since Re d/1-[alpha] > 0 for Re d > 1 - [alpha] and [alpha] < 1, it follows

|t + [alpha]-1+d/1-[alpha]| |[alpha]-1+d/1-[alpha]|[greater than or equal to] and |t| < |t +[alpha]-1+d/1-[alpha]|, [for all] t [greater than or equal to] 0,

and thus

[[phi][.sub.]1](t) [less than or equal to] |1-[alpha]/[alpha]-1+d|, [[phi][.sub.]2](t) < 1, t [greater than or equal to] 0.

Using the above inequalities together with (3.21) we deduce that

|L(z; t)|/|[a[.sub.]1](t)| < r/1-r + r/[(1-r)[.sup.]2,|z|[less than or equal to] r [for all]t [greater than or equal to] 0,

hence the second assumption of Lemma 2.1 holds. Now, from this lemma we obtain that the function L(z; t) is a subordination chain.

Using the fact that (3.7) holds, since G [member of] A, we have that G is convex (univalent) in U. Thus, if we denote by [chi](G(z),zG'(z)) = [phi](z), then L(z; t) = [chi](q(z), tzq'(z)), and the differential equation [chi](G(z), zG'(z)) = [phi](z) has the univalent solution G.

According to Lemma 2.4, we conclude that [empty set](z) <[phi](z) implies G(z) < F(z), and furthermore, since G is a univalent solution of the differential equation X (G(z), zG'(z)) = [empty set](z), it follows that it is the best subordinant of the given differential superordination.

Taking d = 1, s = - 1 and p = 1 in Theorem 3.2, denoting [beta]:= 1 - [alpha], we obtain the next special case:

Corollary 3.2. Let 0 < [beta] < 1 and for a given function g [member of] [SIGMA] suppose that the function [empty set] defined by (3.16) satisfies the condition (3.1), with [delta] given by (3.17).

Let f [member of] [SIGMA] such that [z.sup.2] |[[beta]z[f.sup.1](z) + (1 + [beta])f (z)| is univalent in U and [z.sup.2]f (z) [member of] Q. Then,

[z.sup.2] |[beta]z[g.sup.1](z) | (1 | [beta])g(z)] < [z.sub.2] [[beta]z[f.sup.1](z) | (1 | [beta])f (z)]

implies

[z.sup.2]g(z) X [z.sup.2]f(z),

and the function g is the best subordinant.

Combining the Theorem 3.2 with Theorem 3.1, we obtain the following sandwich-type theorem:

Theorem 3.3. Let [alpha] < 1 and d [member of] C\[mathematical expression not reproducible], with Re d > 1 - [alpha]. For the two given functions [g.sub.1], [g.sub.2] [member of] [[SIGMA].sub.p], suppose that

Re [mathematical expression not reproducible]

where

(3.23) [mathematical expression not reproducible]

and [delta] is given by (3.3).

Let f [member of] [[SIGMA].sub.p] such that [mathematical expression not reproducible] is univalent in U and [mathematical expression not reproducible]. Then,

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

Moreover, the functions [mathematical expression not reproducible] are respectively the best subordinant and the best dominant.

The assumptions that the functions

(3.24) [mathematical expression not reproducible]

and

(3.25) [mathematical expression not reproducible]

need to be univalent in U are difficult to be checked. Thus, in the following sandwich-type result we will replace these assumptions by another sufficient conditions, that are more easy to be verified.

Corollary 3.3. Let [alpha] < 1 and [mathematical expression not reproducible], with Red > 1 - [alpha]. For the given functions f, [g.sub.1], [g.sub.2] [member of] [[SIGMA].sub.p], suppose that

(3.26) [mathematical expression not reproducible]

where [[empty set].sub.1], [[empty set].sub.2] and [[empty set].sub.3] are defined by (3.23) and (3.24) respectively, and [delta] is given by (3.3). Then,

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

Moreover, the functions [mathematical expression not reproducible] (z) and [mathematical expression not reproducible] are respectively the best subordinant and the best dominant.

Proof. In order to prove our corollary, we have to show that the condition (3.26) for k = 3 implies the univalence of the functions [[empty set].sub.3] and [PHI] defined by (3.24) and (3.25).

Since 0 < [delta] [less than or equal to] 1/2 from Remark 3.1, the condition (3.26) for k = 3 means that [[empty set].sub.3] [member of] K(-[delta]) [??] K (-1/2), and from  it follows that [[empty set].sub.3] is a close-to-convex function in U, hence it is univalent in U. Furthermore, by using the same techniques as in the proof of Theorem 3.1 we can prove the convexity (univalence) of [PHI] and so the details may be omitted. Therefore, by applying Theorem 3.3 we obtain the desired result.

The following special case of Corollary 3.3 is obtained for d =1, s = -1 and p =1, with [beta]:= -1 - [alpha]:

Corollary 3.4. Let 0 < [beta] < 1 and for the given functions f, [g.sub.1], [g.sub.2] [member of] [SIGMA], suppose that the inequalities (3.26) hold, where

[[empty set].sub.1](z) = [z.sup.2] [[beta]z[g'.sub.l](z) | (1 + [beta])[g.sub.1](z)], [[empty set].sub.2](z) = [z.sub.2] [[beta]z[g'.sub.2](z) | (1 | [beta][y.sub.2](z)],

[[empty set].sub.3](z) = [z.sup.2] [[beta]zf'(z) | (1 | [beta])f (z)],

and [delta] is given by (3.17). Then,

[z.sup.2] [[beta]z[g'.sub.1][(z) | (1 | [beta])[g.sub.1](z)] < [z.sup.2] [[beta]zf' (z) | (1 | [beta])f (z)] < [z.sup.2] [[beta]z[g'.sub.2](z) | (1 | [beta]Mz)]

implies

[z.sup.2][g.sub.1](z) < [z.sup.2]f (z) < [z.sup.2][g.sub.2](z).

Moreover, the functions [g.sub.1] and [g.sub.2] are respectively the best subordinant and the best dominant.

Next, we will give an interesting special case of our main results, obtained for an appropriate choice of the function g and the corresponding parameters.

Thus, for [alpha] < 1 and [mathematical expression not reproducible], with Re d > 1 - [alpha], let consider the function g [member of] [SIGMA] defined by

[mathematical expression not reproducible]

with

[mathematical expression not reproducible]

where [delta] is given by (3.3), and

[mathematical expression not reproducible]

If the function [empty set] is defined by (3.2) with p =1, then

[empty set](z) = 1 - [(1 + z)sup.-(2[delta]|1)]/ 2[delta]+1,z [member of] U

where the power is the principal one, i.e.

[mathematical expression not reproducible]

A simple computation shows that

[mathematical expression not reproducible]

and from Theorem 3.1 and Theorem 3.2 we obtain:

Example 3.1. Let [alpha] < 1 and d [member of] C \ [mathematical expression not reproducible], with Red > 1 - [alpha], and let [delta] be given by (3.3).

1. If f [member of] [SIGMA] such that

[mathematical expression not reproducible]

then

[mathematical expression not reproducible]

and the right-hand side function is the best dominant (the power is the principal one).

2. If f [member of] [SIGMA] such that [mathematical expression not reproducible] is univalent in U and [mathematical expression not reproducible], then

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

and the right-hand side function is the best subordinant (the power is the principal one).

By similar reasons, for the above mentioned choice of the function g, the Theorem 3.3 reduces to the following sandwich-type results:

Example 3.2. Let [alpha] < 1 and [mathematical expression not reproducible], with Red > 1 - [alpha], and let [[delta].sub.1], [[delta].sub.2] [less than or equal to] [delta] where [delta] is given by (3.3).

If f [member of] [SIGMA] such that [mathematical expression not reproducible] is univalent in U and [mathematical expression not reproducible], then

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

Moreover, the left-hand side functions and the right-hand side are, respectively, the best subordinant and the best dominant (the powers are the principal ones).

For d =1, s = -1 and p = 1, for [beta]:= 1 - [alpha] the Example 3.2 gives us the next result:

Example 3.3. Let 0 < [beta] < 1 and let [[delta].sub.1], [[delta].sub.2] [less than or equal to] [delta] where [delta] is given by (3.17).

If f [member of] [SIGMA] such that [z.sup.2] [[beta]zf'(z) + (1 + [beta])f (z)] is univalent in U and [z.sup.2]f (z) [member of] Q, then

[mathematical expression not reproducible]

implies

[mathematical expression not reproducible]

Moreover, the left-hand side functions and the right-hand side are, respectively, the best subordinant and the best dominant (the powers are the principal ones).

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RABHA M. EL-ASHWAH AND TEODOR BULBOACA

Department of Mathematics, Damietta University, New Damietta 34517, Egypt Faculty of Mathematics and Computer Science, Babes -Bolyai University, 400084 Cluj-Napoca, Romania

E-mail address: r elashwah@yahoo, E-mail address: bulboaca@math.ubbcluj.ro

Key words and phrases. Meromorphic function, convex function, convolution product, differential subordination, differential superordination, integral operator.

Received January, 18, 2018, Accepted December, 3, 2018

2010 Mathematics Subject Classification. 30C80, 30C45.
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