Certain Admissible Classes of Multivalent Functions

Citation for this paper:

Aouf, M.K., Srivastava, H.M., & Seoudy, T.M. (2014). Certain admissible classes of

multivalent functions. Journal of Complex Analysis, Vol. 2014, Article ID 936748.

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Certain Admissible Classes of Multivalent Functions

M.K. Aouf, H.M. Srivastava, & T.M. Seoudy

2014

© 2014 M.K. Aouf et al. This is an open access article distributed under the terms of the Creative Commons Attribution License. http://creativecommons.org/licenses/by/3.0

This article was originally published at:

Research Article

Certain Admissible Classes of Multivalent Functions

M. K. Aouf,

H. M. Srivastava,

and T. M. Seoudy

1_{Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt} 2_{Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4} 3_{Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt} 4_{The University College at Al-Jamoom, Umm Al-Qura University, Makkah, Saudi Arabia}

Correspondence should be addressed to T. M. Seoudy; tms00@fayoum.edu.eg Received 18 July 2014; Accepted 1 September 2014; Published 16 September 2014 Academic Editor: Arcadii Z. Grinshpan

Copyright © 2014 M. K. Aouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate some applications of the differential subordination and the differential superordination of certain admissible classes of multivalent functions in the open unit diskU. Several differential sandwich-type results are also obtained.

1. Introduction

LetH(U) be the class of functions analytic in the open unit disk

U = {𝑧 : 𝑧 ∈ C, |𝑧| < 1} . (1) Denote by H[𝑎, 𝑛] the subclass of H(U) consisting of functions of the form

𝑓 (𝑧) = 𝑎 + 𝑎𝑛𝑧𝑛+ 𝑎𝑛+1𝑧𝑛+1+ ⋅ ⋅ ⋅ (2)

with

H = H [1, 1] . (3) Also let A(𝑝) be the class of all analytic and 𝑝-valent functions of the form

𝑓 (𝑧) = 𝑧𝑝+ ∑∞

𝑛=𝑝+1

𝑎_𝑛𝑧𝑛 (𝑝 ∈ N = {1, 2, 3, . . .} ; 𝑧 ∈ U) . (4) Let𝑓 and 𝐹 be members of the function class H(U). The function𝑓(𝑧) is said to be subordinate to 𝐹(𝑧), or the function 𝐹(𝑧) is said to be superordinate to 𝑓(𝑧), if there exists a function𝜔(𝑧), analytic in U with

𝜔 (0) = 0, |𝜔 (𝑧)| < 1 (𝑧 ∈ U) , (5)

such that

𝑓 (𝑧) = 𝐹 (𝜔 (𝑧)) . (6) In such a case we write𝑓(𝑧) ≺ 𝐹(𝑧). If 𝐹 is univalent in U, then𝑓(𝑧) ≺ 𝐹(𝑧) if and only if 𝑓(0) = 𝐹(0) and 𝑓(U) ⊂ 𝐹(U) (see [1–3]; see also several recent works [4–8] dealing with various properties and applications of the principle of differential subordination and the principle of differential superordination).

We denote byF the set of all functions 𝑞 that are analytic and injective onU \ 𝐸(𝑞), where

𝐸 (𝑞) = {𝜁 ∈ 𝜕U : lim

𝑧 → 𝜁𝑞 (𝑧) = ∞} , (7)

and are such that

𝑞󸀠(𝜁) ̸= 0 (𝜁 ∈ 𝜕U \ 𝐸 (𝑞)) . (8) We further let the subclass of F for which 𝑞(0) = 𝑎 be denoted byF(𝑎) and write

F (1) ≡ F1. (9)

In order to prove our results, we will make use of the following classes of admissible functions.

Definition 1 (see [2, p. 27, Definition 2.3a]). Let Ω be a set inC, 𝑞 ∈ F, and 𝑛 ∈ N. The class Ψ_𝑛[Ω, 𝑞] of admissible

Volume 2014, Article ID 936748, 7 pages http://dx.doi.org/10.1155/2014/936748

2 Journal of Complex Analysis functions consists of those functions𝜓 : C3× U → C that

satisfy the following admissibility condition:

𝜓 (𝑟, 𝑠, 𝑡; 𝑧) ∉ Ω (10) whenever 𝑟 = 𝑞 (𝜁) , 𝑠 = 𝑘𝜁𝑞󸀠(𝜁) , R (𝑡 𝑠+ 1) ≧ 𝑘R (1 + 𝜁𝑞󸀠_(𝜁) 𝑞󸀠_(𝜁)) , (11)

where𝑧 ∈ U, 𝜁 ∈ 𝜕U \ 𝐸(𝑞), and 𝑘 ≧ 𝑛. We write Ψ₁[Ω, 𝑞] simply asΨ[Ω, 𝑞].

In particular, if

𝑞 (𝑧) = (_{𝑀 + 𝑎𝑧}𝑀𝑧 + 𝑎) 𝑀 (𝑀 > 0; |𝑎| < 𝑀) , (12) then

𝑞 (U) = U𝑀= {𝑤 : |𝑤| < 𝑀} , (13)

𝑞(0) = 𝑎, 𝐸(𝑞) = 0, and 𝑞 ∈ F(𝑎). In this case, we set Ψ_𝑛[Ω, 𝑀, 𝑎] = Ψ_𝑛[Ω, 𝑞]. Moreover, in the special case, when we setΩ = U_𝑀, the class is simply denoted byΨ_𝑛[𝑀, 𝑎].

Definition 2 (see [3, p. 817, Definition 3]). Let Ω be a set

in C, 𝑞 ∈ H[𝑎, 𝑛] with 𝑞󸀠(𝑧) ̸= 0. The class Ψ_𝑛󸀠[Ω, 𝑞] of admissible functions consists of those functions𝜓 : C3×U → C that satisfy the following admissibility condition:

𝜓 (𝑟, 𝑠, 𝑡; 𝜁) ∈ Ω (14) whenever 𝑟 = 𝑞 (𝑧) , 𝑠 = 𝑧𝑞󸀠(𝑧) 𝑚 , R (𝑡_𝑠+ 1) ≦_𝑚1R (1 +𝑧𝑞_𝑞󸀠󸀠󸀠(𝑧) (𝑧) ) , (15)

where𝑧 ∈ U, 𝜁 ∈ 𝜕U, and 𝑚 ≧ 𝑛 ≧ 1. In particular, we write Ψ󸀠

1[Ω, 𝑞] simply as Ψ󸀠[Ω, 𝑞].

In our investigation we need the following lemmas which are proved by Miller and Mocanu (see [2] and [3]).

Lemma 3 (see [2, p. 28, Theorem 2.3b]). Let𝜓 ∈ Ψ_𝑛[Ω, 𝑞]

with𝑞(0) = 𝑎. If the analytic function 𝑔(𝑧) given by

𝑔 (𝑧) = 𝑎 + 𝑎𝑛𝑧𝑛+ 𝑎𝑛+1𝑧𝑛+1+ ⋅ ⋅ ⋅ (16)

satisfies the inclusion relationship

𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) ∈ Ω, (17)

then𝑔 ≺ 𝑞.

Lemma 4 (see [3, p. 818, Theorem 1]). Let𝜓 ∈ Ψ_𝑛󸀠[Ω, 𝑞] with 𝑞(0) = 𝑎. If 𝑔 ∈ F(𝑎) and the function

𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) (18)

is univalent inU, then

Ω ⊂ {𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) : 𝑧 ∈ U} (19)

implies that𝑞 ≺ 𝑔.

In this paper, we determine the sufficient conditions for certain admissible classes of multivalent functions so that

𝑞₁(𝑧) ≺ (𝑓 (𝑧)_𝑧𝑝 )𝜇≺ 𝑞₂(𝑧) , (20) where𝜇 > 0 and 𝑞₁and𝑞₂are given univalent functions inU with

𝑞1(0) = 𝑞2(0) = 1. (21)

In addition, we derive several differential sandwich-type results. A similar problem for analytic functions involving certain operators was studied by Aghalary et al. [9], Ali et al. [10], Aouf et al. [11], Kim and Srivastava [12], and other authors (see [13–15]). In particular, unlike the earlier investigation by Aouf and Seoudy [16], we have not used any operators in our present investigation. Nevertheless, for the benefit of the targeted readers of our paper, in addition to oft-cited paper [11], we have included several further citations of recent works (see, e.g., [17–21]) in which various families of linear operators were applied in conjunction with the principle of differential subordination and the principle of differential superordination for the study of analytic or meromorphic multivalent functions.

2. A Set of Subordination Results

Unless otherwise mentioned, we assume throughout this paper that𝑝 ∈ N, 𝜇 > 0, 𝑧 ∈ U, and all power functions are tacitly assumed to denote their principal values.

Definition 5. Let Ω be a set in C and 𝑞 ∈ F₁ ∩ H. The

classΦ[Ω, 𝑞, 𝑝, 𝜇] of admissible functions consists of those functions 𝜙 : C3 × U → C that satisfy the following admissibility condition: 𝜙 (𝑢, V, 𝑤; 𝑧) ∉ Ω (22) whenever 𝑢 = 𝑞 (𝜁) , V = 𝑘𝜁𝑞󸀠(𝜁) + 𝜇𝑝𝑞 (𝜁)_𝜇𝑝 , R (𝑤 − (2𝜇𝑝 − 1) V + 𝜇𝑝𝑢 V − 𝑢 ) ≧ 𝑘R (1 + 𝜁𝑞󸀠󸀠(𝜁) 𝑞󸀠_(𝜁) ) , (23)

where𝑧 ∈ U, 𝜁 ∈ 𝜕U \ 𝐸(𝑞), and 𝑘 ≧ 1. For simplicity, we write

Φ [Ω, 𝑞, 𝑝, 1] = Φ [Ω, 𝑞, 𝑝] . (24)

Theorem 6. Let 𝜙 ∈ Φ[Ω, 𝑞, 𝑝, 𝜇]. If 𝑓 ∈ A(𝑝) satisfies the

condition {𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧)_𝑧𝑝 )𝜇𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧)_𝑧𝑝 )𝜇(𝑧𝑓_{𝑝𝑓(𝑧)}󸀠(𝑧)) 2 ; 𝑧) : 𝑧 ∈ U} ⊂ Ω, (25)

then

(𝑓 (𝑧)_𝑧𝑝 )𝜇≺ 𝑞 (𝑧) . (26)

Proof. We begin by defining the analytic function𝑔 in U by

𝑔 (𝑧) = (𝑓(𝑧)_𝑧𝑝 )𝜇 (𝑧 ∈ U) . (27) Then, in view of (27), we get

(𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧) = 𝑧𝑔󸀠(𝑧) + 𝜇𝑝𝑔 (𝑧) 𝜇𝑝 . (28) Further computations show that

(𝑓(𝑧)_𝑧𝑝 )𝜇𝑧2𝑓󸀠󸀠(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝( 𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓_{𝑝𝑓(𝑧)}󸀠(𝑧)) 2 = 𝑧2𝑔󸀠󸀠(𝑧) + 2𝜇𝑝 𝑧𝑔󸀠(𝑧) + 𝜇𝑝 (𝜇𝑝 − 1) 𝑔 (𝑧)_𝜇𝑝 . (29)

We now define the transformations fromC3toC by 𝑢 = 𝑟, V =𝑠 + 𝜇𝑝𝑟

𝜇𝑝 , 𝑤 = 𝑡 + 2𝜇𝑝𝑠 + 𝜇𝑝 (𝜇𝑝 − 1) 𝑟

𝜇𝑝

(30)

and suppose that

𝜓 (𝑟, 𝑠, 𝑡; 𝑧) = 𝜙 (𝑢, V, 𝑤; 𝑧) = 𝜙 (𝑟,𝑠 + 𝜇𝑝𝑟 𝜇𝑝 , 𝑡 + 2𝜇𝑝𝑠 + 𝜇𝑝 (𝜇𝑝 − 1) 𝑟 𝜇𝑝 ; 𝑧) . (31) The proof will make use ofLemma 3. Indeed, by using (27) to (31), we obtain 𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) = 𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), (𝑓 (𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓 (𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧)) 2 ; 𝑧) . (32) Hence (25) becomes 𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) ∈ Ω. (33) The proof is completed if it can be shown that the admis-sibility condition for𝜙 ∈ Φ[Ω, 𝑞, 𝑝, 𝜇] is equivalent to the

admissibility condition for𝜓 as given inDefinition 1. We note that

𝑡 𝑠+ 1 =

𝑤 − (2𝜇𝑝 − 1) V + 𝜇𝑝𝑢

V − 𝑢 , (34) and hence𝜓 ∈ Ψ₁[Ω, 𝑞]. ByLemma 3, we thus obtain

𝑔 (𝑧) ≺ 𝑞 (𝑧) or (𝑓(𝑧)_𝑧𝑝 )𝜇≺ 𝑞 (𝑧) . (35) which evidently provesTheorem 6.

If Ω ̸= C is a simply connected domain, then Ω = ℎ(U) for some conformal mapping ℎ of U onto Ω. In this case, the class Φ[ℎ(U), 𝑞, 𝑝, 𝜇] is written, for convenience, asΦ[ℎ, 𝑞, 𝑝, 𝜇]. The following result is an immediate conse-quence ofTheorem 6.

Theorem 7. Let 𝜙 ∈ Φ[ℎ, 𝑞, 𝑝, 𝜇]. If 𝑓 ∈ A(𝑝) satisfies the

condition, 𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) ≺ ℎ (𝑧) , (36) then (𝑓(𝑧)_𝑧𝑝 )𝜇≺ 𝑞 (𝑧) . (37) Putting 𝜇 = 1 in Theorem 7, we obtain the following corollary.

Corollary 8. Let 𝜙 ∈ Φ[ℎ, 𝑞, 𝑝]. If 𝑓 ∈ A(𝑝) satisfies the

condition 𝜙 (𝑓 (𝑧) 𝑧𝑝 , 𝑓󸀠_(𝑧) 𝑝𝑧𝑝−1, 𝑓󸀠󸀠_(𝑧) 𝑝𝑧𝑝−2) ≺ ℎ (𝑧) , (38) then 𝑓 (𝑧) 𝑧𝑝 ≺ 𝑞 (𝑧) . (39)

Our next result is an extension ofTheorem 6to the case where the behavior of𝑞 on 𝜕U is not known.

Corollary 9. Let Ω ⊂ C and suppose that the function 𝑞 is

univalent inU with 𝑞(0) = 1. Also let 𝜙 ∈ Φ[Ω, 𝑞_𝜌, 𝑝, 𝜇] for some𝜌 ∈ (0, 1), where 𝑞_𝜌(𝑧) = 𝑞(𝜌𝑧). If 𝑓 ∈ A(𝑝) and

𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) ∈ Ω, (40) then (𝑓(𝑧) 𝑧𝑝 ) 𝜇 ≺ 𝑞 (𝑧) . (41)

4 Journal of Complex Analysis

Proof. Theorem 6readily yields

(𝑓(𝑧) 𝑧𝑝 )

𝜇

≺ 𝑞_𝜌(𝑧) . (42) The asserted result is now deduced from the fact that𝑞_𝜌(𝑧) ≺ 𝑞(𝑧).

Theorem 10. Let the functions ℎ and 𝑞 be univalent in U, with

𝑞(0) = 1, and set

𝑞_𝜌(𝑧) = 𝑞 (𝜌𝑧) , ℎ_𝜌(𝑧) = ℎ (𝜌𝑧) . (43)

Also let𝜙 : C3 × U → C satisfy one of the following

conditions:

(1)𝜙 ∈ Φ[ℎ, 𝑞_𝜌, 𝑝, 𝜇] for some 𝜌 ∈ (0, 1) or

(2) there exists𝜌₀∈ (0, 1) such that 𝜙 ∈ Φ[ℎ_𝜌, 𝑞_𝜌, 𝑝, 𝜇] for

all𝜌 ∈ (𝜌₀, 1).

If𝑓 ∈ A(𝑝) satisfies condition (36), then (𝑓(𝑧)

𝑧𝑝 ) 𝜇

≺ 𝑞 (𝑧) . (44)

Proof. The proof of Theorem 10 is similar to the proof of

a known result [2, p. 30, Theorem 2.3d] and is, therefore, omitted.

The next theorem yields the best dominant of differential subordination (36).

Theorem 11. Let the function ℎ be univalent in U. Also let 𝜙 :

C3× U → C. Suppose that the differential equation

𝜙 (𝑞 (𝑧) , 𝑧𝑞󸀠(𝑧) , 𝑧2𝑞󸀠󸀠(𝑧) ; 𝑧) = ℎ (𝑧) (45)

has a solution𝑞 with 𝑞(0) = 1 and satisfies one of the following conditions:

(1)𝑞 ∈ F₁and𝜙 ∈ Φ[ℎ, 𝑞, 𝑝, 𝜇];

(2) the function𝑞 is univalent in U and 𝜙 ∈ Φ[ℎ, 𝑞_𝜌, 𝑝, 𝜇]

for some𝜌 ∈ (0, 1); or

(3) the function𝑞 is univalent in U and there exists 𝜌₀ ∈ (0, 1) such that 𝜙 ∈ Φ[ℎ_𝜌, 𝑞_𝜌, 𝑝, 𝜇] for all 𝜌 ∈ (𝜌₀, 1). If 𝑓 ∈ A(𝑝) satisfies (36), then

(𝑓(𝑧) 𝑧𝑝 )

𝜇

≺ 𝑞 (𝑧) (46)

and𝑞 is the best dominant.

Proof. Following the same arguments in [2, p. 31, Theorem

2.3e], we deduce that𝑞 is a dominant from Theorems7and10. Since𝑞 satisfies (45), it is also a solution of (36) and, therefore, 𝑞 will be dominated by all dominants. Hence 𝑞 is the best dominant.

In the particular case when𝑞(𝑧) = 1 + 𝑀𝑧 (𝑀 > 0), in view ofDefinition 5, the classΦ[Ω, 𝑞, 𝑝, 𝜇] of admissible functions, denoted byΦ[Ω, 𝑀, 𝑝, 𝜇], is described below.

Definition 12. Let Ω be a set in C and 𝑀 > 0. The

classΦ[Ω, 𝑀, 𝑝, 𝜇] of admissible functions consists of those functions𝜙 : C3× U → C such that

𝜙 (1 + 𝑀𝑒𝑖𝜃, 1 +𝑘 + 𝜇𝑝 𝜇𝑝 𝑀𝑒𝑖𝜃, 𝐿 + 𝜇𝑝 [(2𝑘 + 𝜇𝑝 − 1) 𝑀𝑒𝑖𝜃_{+ 𝜇𝑝 − 1]} 𝜇𝑝 ; 𝑧) ∉ Ω (47) whenever𝑧 ∈ U, 𝜃 ∈ R, and R (𝐿𝑒−𝑖𝜃) ≧ (𝑘 − 1) 𝑘𝑀 (48) for all real𝜃 and 𝑘 ≧ 𝜇𝑝.

Corollary 13. Let 𝜙 ∈ Φ[Ω, 𝑀, 𝑝, 𝜇]. If 𝑓 ∈ A(𝑝) satisfies the

condition 𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓 (𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓 (𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) ∈ Ω, (49) then 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨(𝑓(𝑧)𝑧𝑝 ) 𝜇 − 1󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨 < 𝑀 (𝑧 ∈ U).} (50) In the special case when

Ω = 𝑞 (U) = {𝜔 : |𝜔 − 1| < 𝑀} , (51) the classΦ[Ω, 𝑀, 𝑝, 𝜇] is simply denoted by Φ[𝑀, 𝑝, 𝜇].

Corollary 14. Let 𝜙 ∈ Φ[𝑀]. If 𝑓 ∈ A(𝑝) satisfies the

condition 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨𝜙 (( 𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) − 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 󵄨󵄨< 𝑀, (52) then 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨(𝑓 (𝑧)𝑧𝑝 ) 𝜇 − 1󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨 < 𝑀.} (53)

Corollary 15. If 𝑘 ≧ 1 and 𝑓 ∈ A(𝑝) satisfies the condition

󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 󵄨( 𝑓 (𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧) − 1󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨_󵄨< 𝑀, (54) then 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨(𝑓 (𝑧)𝑧𝑝 ) 𝜇 − 1󵄨󵄨󵄨󵄨_{󵄨󵄨󵄨󵄨 < 𝑀.} (55)

Proof. Corollary 15follows fromCorollary 14upon setting

3. Superordination and Sandwich-Type Results

In this section we investigate the dual problem of differential subordination, that is, differential superordination of mul-tivalent functions. For this purpose, the class of admissible functions is given in the following definition.

Definition 16. LetΩ be a set in C and 𝑞 ∈ H with 𝑧𝑞󸀠(𝑧) ̸=

0. The class Φ󸀠_{[Ω, 𝑞, 𝑝, 𝜇] of admissible functions consists of}

those functions𝜙 : C3× U → C that satisfy the following admissibility condition: 𝜙 (𝑢, V, 𝑤; 𝜁) ∈ Ω (57) whenever 𝑢 = 𝑞 (𝑧) , V = 𝑧𝑞󸀠(𝑧) + 𝑚𝜇𝑝𝑞 (𝑧) 𝑚𝜇𝑝 , R (𝑤 − (2𝜇𝑝 − 1) V + 𝜇𝑝𝑢 V − 𝑢 ) ≧ 1 𝑚R (1 + 𝜁𝑞󸀠󸀠_(𝜁) 𝑞󸀠_(𝜁) ) , (58) where𝑧 ∈ U, 𝜁 ∈ 𝜕U, and 𝑚 ≧ 1. For convenience, we write

Φ󸀠[Ω, 𝑞, 𝑝, 1] = Φ󸀠[Ω, 𝑞, 𝑝] . (59)

Theorem 17. Let 𝜙 ∈ Φ󸀠_{[Ω, 𝑞, 𝑝, 𝜇]. If 𝑓 ∈ A(𝑝),}

(𝑓(𝑧) 𝑧𝑝 ) 𝜇 ∈ F₁ 𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) (60) is univalent inU, then Ω ⊂ {𝜙 ((𝑓(𝑧) 𝑧𝑝 ) 𝜇 , (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) : 𝑧 ∈ U} (61) implies that 𝑞 (𝑧) ≺ (𝑓(𝑧)_𝑧𝑝 )𝜇. (62)

Proof. From (32) and (61), we find that

Ω ⊂ {𝜓 (𝑔 (𝑧) , 𝑧𝑔󸀠(𝑧) , 𝑧2𝑔󸀠󸀠(𝑧) ; 𝑧) : 𝑧 ∈ U} . (63) We also see from (30) that the admissibility condition for the function class 𝜙 ∈ Φ󸀠[Ω, 𝑞, 𝑝, 𝜇] is equivalent to the

admissibility condition for𝜓 as given inDefinition 2. Hence 𝜓 ∈ Ψ󸀠

1[Ω, 𝑞]. Thus, byLemma 4, we have

𝑞 (𝑧) ≺ 𝑔 (𝑧) or 𝑞 (𝑧) ≺ (𝑓(𝑧)_𝑧𝑝 )𝜇, (64) which evidently completes the proof ofTheorem 17.

IfΩ ̸= C is a simply connected domain, then Ω = ℎ(U) for some conformal mappingℎ of U onto Ω. In this case, the classΦ󸀠[ℎ(U), 𝑞, 𝑝, 𝜇] is written simply as Φ󸀠[ℎ, 𝑞, 𝑝, 𝜇].

Proceeding similarly as inSection 2, the following result can be derived as an immediate consequence ofTheorem 17.

Theorem 18. Let the function ℎ be analytic in U and 𝜙 ∈

Φ󸀠_{[ℎ, 𝑞, 𝑝, 𝜇]. If 𝑓 ∈ A(𝑝),} (𝑓(𝑧)_𝑧𝑝 )𝜇∈ F1, 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧)_𝑧𝑝 )𝜇𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧)_𝑧𝑝 )𝜇(𝑧𝑓_{𝑝𝑓(𝑧)}󸀠(𝑧)) 2 ; 𝑧) (65) is univalent inU, then ℎ (𝑧) ≺ 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) (66) implies that 𝑞 (𝑧) ≺ (𝑓(𝑧)_𝑧𝑝 )𝜇. (67) Putting 𝜇 = 1 in Theorem 18, we obtain the following corollary.

Corollary 19. Let the function ℎ be analytic in U and 𝜙 ∈

Φ󸀠_{[ℎ, 𝑞, 𝑝]. If 𝑓 ∈ A(𝑝),} 𝑓 (𝑧) 𝑧𝑝 ∈ F1, 𝜙 (𝑓 (𝑧)_𝑧𝑝 ,_𝑝𝑧𝑓󸀠(𝑧)_𝑝−1,𝑓_𝑝𝑧󸀠󸀠_𝑝−2(𝑧); 𝑧) (68) is univalent inU, then ℎ (𝑧) ≺ 𝜙 (𝑓 (𝑧)_𝑧𝑝 ,𝑓󸀠(𝑧) 𝑝𝑧𝑝−1, 𝑓󸀠󸀠_(𝑧) 𝑝𝑧𝑝−2; 𝑧) (69) implies that 𝑞 (𝑧) ≺ 𝑓 (𝑧)_𝑧𝑝 . (70)

6 Journal of Complex Analysis Theorems17and18can only be used to obtain

subordi-nants of the differential superordination of the form (61) or (66). The following theorem proves the existence of the best subordinant of (66) for a specified𝜙.

Theorem 20. Let the function ℎ be analytic in U and 𝜙 : C3_×

U → C. Suppose that the differential equation

𝜙 (𝑞 (𝑧) , 𝑧𝑞󸀠_{(𝑧) , 𝑧}2_𝑞󸀠󸀠_{(𝑧) ; 𝑧) = ℎ (𝑧) (71)}

has a solution𝑞 ∈ F₁. If𝜙 ∈ Φ󸀠[ℎ, 𝑞, 𝑝, 𝜇],𝑓 ∈ A(𝑝),

(𝑓(𝑧)_𝑧𝑝 )𝜇∈ F₁ 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧)_𝑧𝑝 )𝜇𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧)_𝑧𝑝 )𝜇(𝑧𝑓_{𝑝𝑓(𝑧)}󸀠(𝑧)) 2 ; 𝑧) (72) is univalent inU, then ℎ (𝑧) ≺ 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧𝑓󸀠_(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧) 𝑧𝑝 ) 𝜇 (𝑧𝑓󸀠(𝑧) 𝑝𝑓(𝑧)) 2 ; 𝑧) (73) implies that 𝑞 (𝑧) ≺ (𝑓(𝑧)_𝑧𝑝 )𝜇 (74)

and𝑞 is the best subordinant.

Proof. The proof is similar to the proof ofTheorem 11. We,

therefore, omit the details involved.

Combining Theorems7and18, we obtain the following sandwich-type theorem.

Theorem 21. Let the functions ℎ1and𝑞1be analytic inU, the

functionℎ₂univalent inU, 𝑞₂∈ F₁with

𝑞₁(0) = 𝑞₂(0) = 1, 𝜙 ∈ Φ [ℎ₂, 𝑞₂, 𝑝, 𝜇] ∩ Φ󸀠[ℎ₁, 𝑞₁, 𝑝, 𝜇] . (75) If𝑓 ∈ A(𝑝), (𝑓(𝑧) 𝑧𝑝 ) 𝜇 ∈ H ∩ F₁, 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧)_𝑧𝑝 )𝜇𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓(𝑧)_𝑧𝑝 )𝜇(𝑧𝑓_{𝑝𝑓(𝑧)}󸀠(𝑧)) 2 ; 𝑧) (76) is univalent inU, then ℎ₁(𝑧) ≺ 𝜙 ((𝑓(𝑧)_𝑧𝑝 )𝜇, (𝑓(𝑧)_𝑧𝑝 )𝜇𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧), ( 𝑓(𝑧) 𝑧𝑝 ) 𝜇_𝑧2_𝑓󸀠󸀠_(𝑧) 𝑝𝑓 (𝑧) + (𝜇 − 1) 𝑝(𝑓 (𝑧)_𝑧𝑝 )𝜇(𝑧𝑓󸀠(𝑧) 𝑝𝑓 (𝑧)) 2 ; 𝑧) ≺ ℎ₂(𝑧) (77) implies that 𝑞₁(𝑧) ≺ (𝑓 (𝑧)_𝑧𝑝 )𝜇≺ 𝑞₂(𝑧) . (78) Upon setting𝜇 = 1 inTheorem 21, we get the following result.

Corollary 22. Let the functions ℎ₁and𝑞₁be analytic inU, the functionℎ₂univalent inU, 𝑞₂∈ F₁with

𝑞₁(0) = 𝑞2(0) = 1, (79) and𝜙 ∈ Φ[ℎ₂, 𝑞₂, 𝑝] ∩ Φ󸀠[ℎ₁, 𝑞₁, 𝑝]. If 𝑓 ∈ A(𝑝), 𝑓 (𝑧) 𝑧𝑝 ∈ H ∩ F1, 𝜙 (𝑓 (𝑧)_𝑧𝑝 ,_𝑝𝑧𝑓󸀠(𝑧)_𝑝−1,𝑓_𝑝𝑧󸀠󸀠_𝑝−2(𝑧)) (80) is univalent inU, then ℎ₁(𝑧) ≺ 𝜙 (𝑓 (𝑧)_𝑧𝑝 ,𝑓󸀠(𝑧) 𝑝𝑧𝑝−1, 𝑓󸀠󸀠_(𝑧) 𝑝𝑧𝑝−2) ≺ ℎ2(𝑧) (81) implies that 𝑞₁(𝑧) ≺𝑓 (𝑧)_𝑧𝑝 ≺ 𝑞₂(𝑧) . (82)

Conflict of Interests

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

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