Subordination and Superordination on Schwarzian Derivatives

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 712328, 18 pages doi:10.1155/2008/712328

Research Article Subordination and Superordination on Schwarzian Derivatives Rosihan M. Ali,1 V. Ravichandran,2 and N. Seenivasagan3 1

School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia Department of Mathematics, University of Delhi, Delhi 110 007, India 3 Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613 005, India 2

Correspondence should be addressed to Rosihan M. Ali, [email protected] Received 4 September 2008; Accepted 30 October 2008 Recommended by Paolo Ricci Let the functions q1 be analytic and let q2 be analytic univalent in the unit disk. Using the methods of differential subordination and superordination, sufficient conditions involving the Schwarzian derivative of a normalized analytic function f are obtained so that either q1 z ≺ zf z/fz ≺ q2 z or q1 z ≺ 1 zf z/f z ≺ q2 z. As applications, sufficient conditions are determined relating the Schwarzian derivative to the starlikeness or convexity of f. Copyright q 2008 Rosihan M. Ali 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.

1. Introduction Let HU be the class of functions analytic in U : {z ∈ C : |z| < 1} and Ha, n be the subclass of HU consisting of functions of the form fz a an zn an1 zn1 · · · . We will write H ≡ H1, 1 . Denote by A the subclass of H0, 1 consisting of normalized functions f of the form fz z

∞ ak zk

z ∈ U.

1.1

k2

Let S∗ and K, respectively, be the familiar subclasses of A consisting of starlike and convex functions in U. The Schwarzian derivative {f, z} of an analytic, locally univalent function f is defined by {f, z} :

f z f z

2 1 f z − . 2 f z

1.2

2

Journal of Inequalities and Applications

Owa and Obradović 1 proved that if f ∈ A satisfies R

zf z 2 1 z2 {f, z} > 0, 1 2 f z

1.3

then f ∈ K. Miller and Mocanu 2 proved that if f ∈ A satisfies one of the following conditions:

zf z 2 R 1 αz {f, z} > 0 Rα ≥ 0, f z zf z 2 R 1 z2 {f, z} > 0, f z

1.4

or R

1

zf z z2 {f,z} e > 0, f z

1.5

then f ∈ K. In fact, Miller and Mocanu 2 found conditions on φ : C2 × U → C such that zf z 2 , z {f, z}; z R φ 1 >0 f z

1.6

implies f ∈ K. Each of the conditions mentioned above readily followed by choosing an appropriate φ. Miller and Mocanu 2 also found conditions on φ : C3 × U → C such that zf z 2 zf z ,1 , z {f, z}; z R φ >0 fz f z

1.7

implies f ∈ S∗ . As applications, if f ∈ A satisfies either zf z zf z 2 zf z β 1 z {f, z} > 0 R α fz f z fz

α, β ∈ R,

1.8

or R

zf z fz

zf z 1 1 z2 {f, z} >− , f z 2

1.9

then f ∈ S∗ . Let f and F be members of HU. The function f is said to be subordinate to F, or F is said to be superordinate to f, written fz ≺ Fz, if there exists a function w analytic in U with w0 0 and |wz| < 1 z ∈ U, such that fz Fwz. If F is univalent, then fz ≺ Fz if and only if f0 F0 and fU ⊂ FU.

Rosihan M. Ali et al.

3

In this paper, sufficient conditions involving the Schwarzian derivatives are obtained for functions f ∈ A to satisfy either q1 z ≺

zf z ≺ q2 z fz

or q1 z ≺ 1

zf z ≺ q2 z, f z

1.10

where the functions q1 are analytic and q2 is analytic univalent in U. In Section 2, a class of admissible functions is introduced. Sufficient conditions on functions f ∈ A are obtained so that zf z/fz is subordinated to a given analytic univalent function q in U. As a consequence, we obtained the result 1.7 of Miller and Mocanu 2 relating the Schwarzian derivatives to the starlikeness of functions f ∈ A. Recently, Miller and Mocanu 3 investigated certain first- and second-order differential superordinations, which is the dual problem to subordination. Several authors have continued the investigation on superordination to obtain sandwich-type results 4–20 . In Section 3, superordination is investigated on a class of admissible functions. Sufficient conditions involving the Schwarzian derivatives of functions f ∈ A are obtained so that zf z/fz is superordinated to a given analytic subordinant q in U. For q1 analytic and q2 analytic univalent in U, sandwich-type results of the form q1 z ≺

zf z ≺ q2 z fz

1.11

are obtained. This result extends earlier works by several authors. Section 4 is devoted to finding sufficient conditions for functions f ∈ A to satisfy q1 z ≺ 1

zf z ≺ q2 z. f z

1.12

As a consequence, we obtained the result 1.6 of Miller and Mocanu 2 . To state our results, we need the following preliminaries. Denote by Q the set of all functions q that are analytic and injective on U \ Eq, where

Eq ζ ∈ ∂U : lim qz ∞ , z→ζ

1.13

0 for ζ ∈ ∂U \ Eq. Further, let the subclass of Q for which q0 a and are such that q ζ / be denoted by Qa and Q1 ≡ Q1 . Definition 1.1 see 2, Definition 2.3a, page 27 . Let Ω be a set in C, q ∈ Q and let n be a positive integer. The class of admissible functions Ψn Ω, q consists of those functions ψ : C3 × U → C that satisfy the admissibility condition ψr, s, t; z / ∈Ω

1.14

4


whenever r qζ, s kζq ζ, and R

ζq ζ t 1 ≥ kR 1 , s q ζ

1.15

z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ n. We write Ψ1 Ω, q as ΨΩ, q . If ψ : C2 × U → C, then the admissibility condition 1.14 reduces to ∈ Ω, ψqζ, kζq ζ; z /

1.16

z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ n. Definition 1.2 see 3, Definition 3, page 817 . Let Ω be a set in C, q ∈ Ha, n with q z / 0. The class of admissible functions Ψn Ω, q consists of those functions ψ : C3 × U → C that satisfy the admissibility condition ψr, s, t; ζ ∈ Ω

1.17

whenever r qz, s zq z/m, and R

zq z t 1 1 ≤ R 1 , s m q z

1.18

z ∈ U, ζ ∈ ∂U, and m ≥ n ≥ 1. In particular, we write Ψ1 Ω, q as Ψ Ω, q . If ψ : C2 × U → C, then the admissibility condition 1.17 reduces to zq z ψ qz, ; ζ ∈ Ω, m

1.19

z ∈ U, ζ ∈ ∂U and m ≥ n. Lemma 1.3 see 2, Theorem 2.3b, page 28 . Let ψ ∈ Ψn Ω, q with q0 a. If the analytic function pz a an zn an1 zn1 · · · satisfies ψ pz, zp z, z2 p z; z ∈ Ω,

1.20

then pz ≺ qz. Lemma 1.4 see 3, Theorem 1, page 818 . Let ψ ∈ Ψn Ω, q with q0 a. If p ∈ Qa and ψpz, zp z, z2 p z; z is univalent in U, then

Ω ⊂ ψpz, zp z, z2 p z; z : z ∈ U implies qz ≺ pz.

1.21


5

2. Subordination and starlikeness We first define the following class of admissible functions that are required in our first result. Definition 2.1. Let Ω be a set in C and q ∈ Q1 . The class of admissible functions ΦS Ω, q consists of those functions φ : C3 × U → C that satisfy the admissibility condition φu, v, w; z / ∈Ω

2.1

kζq ζ qζ / 0, qζ ζq ζ 2w u2 − 1 3v − u2 1 , R ≥ kR 2v − u q ζ

2.2

whenever u qζ,

v qζ

z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1. Theorem 2.2. Let f ∈ A with fzf z/z / 0. If φ ∈ ΦS Ω, q and zf z 2 zf z ,1 , z {f, z}; z : z ∈ U ⊂ Ω, φ fz f z

2.3

zf z ≺ qz. fz

2.4

zf z . fz

2.5

then

Proof. Define the function p by pz :

A simple calculation yields 1

zf z zp z pz . f z pz

2.6

Further computations show that zp z z2 p z 3 zp z 2 1 − p2 z z {f, z} − . pz 2 pz 2 2

2.7

6


Define the transformation from C3 to C3 by s vr , r

u r,

s t 3 s 2 1 − r2 w − . r 2 r 2

2.8

Let s s t 3 s 2 1 − r2 ψr, s, t; z φu, v, w; z φ r, r , − ;z . r r 2 r 2

2.9

The proof will make use of Lemma 1.3. Using 2.5, 2.6, and 2.7, from 2.9 we obtain zf z 2 zf z ,1 , z {f, z}; z . ψ pz, zp z, z p z; z φ fz f z

2

2.10

Hence 2.3 becomes ψ pz, zp z, z2 p z; z ∈ Ω.

2.11

A computation using 2.8 yields t 2w u2 − 1 3v − u2 1 . s 2v − u

2.12

Thus the admissibility condition for φ ∈ ΦS Ω, q in Definition 2.1 is equivalent to the admissibility condition for ψ as given in Definition 1.1. Hence ψ ∈ ΨΩ, q and by Lemma 1.3, pz ≺ qz or zf z ≺ qz. fz

2.13

If Ω / C is a simply connected domain, then Ω hU for some conformal mapping h of U onto Ω. In this case, the class ΦS hU, q is written as ΦS h, q . The following result is an immediate consequence of Theorem 2.2. Theorem 2.3. Let φ ∈ ΦS h, q . If f ∈ A with fzf z/z / 0 satisfies φ

zf z 2 zf z ,1 , z {f, z}; z ≺ hz, fz f z

2.14

then zf z ≺ qz. fz

2.15


7

Following similar arguments as in 2, Theorem 2.3d, page 30 , Theorem 2.3 can be extended to the following theorem where the behavior of q on ∂U is not known. Theorem 2.4. Let h and q be univalent in U with q0 1, and set qρ z qρz and hρ z hρz. Let φ : C3 × U → C satisfy one of the following conditions: i φ ∈ ΦS h, qρ for some ρ ∈ 0, 1, or ii there exists ρ0 ∈ 0, 1 such that φ ∈ ΦS hρ , qρ for all ρ ∈ ρ0 , 1. If f ∈ A with fzf z/z / 0 satisfies 2.14, then zf z ≺ qz. fz

2.16

The next theorem yields the best dominant of the differential subordination 2.14. Theorem 2.5. Let h be univalent in U, and φ : C3 × U → C. Suppose that the differential equation zq z zq z z2 q z 3 zq z 2 1 − q2 z , − ; z hz φ qz, qz qz qz 2 qz 2

2.17

has a solution q with q0 1 and one of the following conditions is satisfied: 1 q ∈ Q1 and φ ∈ ΦS h, q , 2 q is univalent in U and φ ∈ ΦS h, qρ for some ρ ∈ 0, 1, or 3 q is univalent in U and there exists ρ0 ∈ 0, 1 such that φ ∈ ΦS hρ , qρ for all ρ ∈ ρ0 , 1. If f ∈ A with fzf z/z / 0 satisfies 2.14, then zf z ≺ qz, fz

2.18

and q is the best dominant. Proof. Applying the same arguments as in 2, Theorem 2.3e, page 31 , we first note that q is a dominant from Theorems 2.3 and 2.4. Since q satisfies 2.17, it is also a solution of 2.14, and therefore q will be dominated by all dominants. Hence q is the best dominant. We will apply Theorem 2.2 to two specific cases. First, let qz 1 Mz, M > 0. Theorem 2.6. Let Ω be a set in C, and φ : C3 × U → C satisfy the admissibility condition kMeiθ iθ iθ ∈Ω , L; z / φ 1 Me , 1 Me 1 Meiθ

2.19

8


whenever z ∈ U, θ ∈ R, with R

2 3k2 M2 2L 1 Meiθ − 1 e−iθ M −iθ e M

≥ 2k2 M

2.20

for all real θ and k ≥ 1. If f ∈ A with fzf z/z / 0 satisfies

zf z zf z 2 φ ,1 , z {f, z}; z ∈ Ω, fz f z

2.21

zf z fz − 1 < M.

2.22

then

Proof. Let qz 1 Mz, M > 0. A computation shows that the conditions on φ implies that it belongs to the class of admissible functions ΦS Ω, 1 Mz . The result follows immediately from Theorem 2.2. In the special case Ω qU {ω : |ω − 1| < M}, the conclusion of Theorem 2.6 can be written as φ zf z , 1 zf z , z2 {f, z}; z − 1 < M ⇒ zf z − 1 < M. fz f z fz

2.23

Example 2.7. The functions φ1 u, v, w; z : 1 − αu αv, α ≥ 2M − 1 ≥ 0 and φ2 u, v, w; z : v/u, 0 < M ≤ 2 satisfy the admissibility condition 2.19 and hence Theorem 2.6 yields 1 − α zf z α 1 zf z − 1 < M ⇒ zf z − 1 < M fz f z fz 1 zf z/f z < M ⇒ zf z − 1 < M − 1 zf z/fz fz

α ≥ 2M − 1 ≥ 0, 0 < M ≤ 2. 2.24

By considering the function φu, v, w; z : uv−1λu−1 with 0 < M ≤ 1, λ2−M ≥ 0, it follows again from Theorem 2.6 that 2 zf z z f z zf z fz λ fz − 1 ≤ M2 λ − M ⇒ fz − 1 < M.

2.25

This above implication was obtained in 21, Corollary 2, page 583 . A second application of Theorem 2.2 is to the case qU being the half-plane qU {w : Rw > 0} : Δ.


9

Theorem 2.8. Let Ω be a set in C and let the function φ : C3 × U → C satisfy the admissibility condition φiρ, iτ, ξ iη; z / ∈Ω

2.26

for all z ∈ U and for all real ρ, τ, ξ and η with ρτ ≥

1 1 3ρ2 , 2

ρη ≥ 0.

2.27

Let f ∈ A with f zfz/z / 0. If zf z 2 zf z ,1 , z {f, z}; z ∈ Ω, φ fz f z

2.28

then f ∈ S∗ . Proof. Let qz : 1 z/1 − z; then q0 1, Eq {1} and q ∈ Q1 . For ζ : eiθ ∈ ∂U \ {1}, we obtain ζq ζ −

qζ iρ,

1 ρ2 , 2

ζ2 q ζ

1 ρ2 1 − iρ , 2

2.29

where ρ : cotθ/2. Note that R

ζq ζ 1 0 ζ / 1. q ζ

2.30

We next describe the class of admissible functions ΦS Ω, 1 z/1 − z in Definition 2.1. For ζ / 1, u qζ : iρ,

v qζ

k1 ρ2 kζq ζ i ρ : iτ, qζ 2ρ

w ξ iη

2.31

with R

2w u2 − 1 3v − u2 2v − u

2ρη . k1 ρ2

2.32

Thus the admissibility condition for functions in ΦS Ω, 1 z/1 − z is equivalent to 2.26, whence φ ∈ ΦS Ω, 1 z/1 − z . From Theorem 2.2, we deduce that f ∈ S∗ . When hz 1 z/1 − z, then hU Δ qU. Writing the class of admissible functions ΦS hU, Δ as ΦS Δ , the following result is a restatement of 1.7, which is an immediate consequence of Theorem 2.8.

10


0 Corollary 2.9 see 2, Theorem 4.6a, page 244 . Let φ ∈ ΦS Δ . If f ∈ A with fzf z/z / satisfies zf z 2 zf z ,1 , z {f, z}; z > 0, R φ fz f z

2.33

then f ∈ S∗ . 3. Superordination and starlikeness Now we will give the dual result of Theorem 2.2 for differential superordination. Definition 3.1. Let Ω be a set in C, q ∈ H with zq z / 0. The class of admissible functions 3 ΦS Ω, q consists of those functions φ : C × U → C that satisfy the admissibility condition φu, v, w; ζ ∈ Ω

3.1

whenever u qz,

v qz

zq z mqz

2w u2 − 1 3v − u2 R 2v − u

qz /0 , / 0, zq z

zq z 1 1 , ≤ R m q z

3.2

z ∈ U, ζ ∈ ∂U and m ≥ 1. Theorem 3.2. Let φ ∈ ΦS Ω, q , and f ∈ A with f zfz/z / 0. If zf z/fz ∈ Q1 and φzf z/fz, 1 zf z/f z, z2 {f, z}; z is univalent in U, then zf z 2 zf z ,1 , z {f, z}; z : z ∈ U Ω⊂ φ fz f z

3.3

implies qz ≺

zf z . fz

3.4

Proof. With pz zf z/fz, and r s s t 3 s 2 1 − r2 , ; z φu, v, w; z, ψr, s, t; z φ r, r r 2 r 2

3.5

equations 2.10 and 3.3 yield

Ω ⊂ ψ pz, zp z, z2 p z; z : z ∈ U .

3.6


11

Since 2w u2 − 1 3v − u2 t 1 , s 2v − u

3.7

the admissibility condition for φ ∈ ΦS Ω, q is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence ψ ∈ Ψ Ω, q , and by Lemma 1.4, qz ≺ pz or qz ≺

zf z . fz

3.8

If Ω / C is a simply connected domain, then Ω hU for some conformal mapping h of U onto Ω. With ΦS hU, q as ΦS h, q , Theorem 3.2 can be written in the following form. Theorem 3.3. Let q ∈ H, h be analytic in U and φ ∈ ΦS h, q . If f ∈ A, f zfz/z / 0, zf z/fz ∈ Q1 and φzf z/fz, 1 zf z/f z, z2 {f, z}; z is univalent in U, then hz ≺ φ

zf z 2 zf z ,1 , z {f, z}; z fz f z

3.9

implies qz ≺

zf z . fz

3.10

Theorems 3.2 and 3.3 can only be used to obtain subordinants of differential superordinations of the form 3.3 or 3.9. The following theorem proves the existence of the best subordinant of 3.9 for an appropriate φ. Theorem 3.4. Let h be analytic in U and φ : C3 × U → C. Suppose that the differential equation

zq z zq z z2 q z 3 zq z 2 1 − q2 z φ qz, qz , − ; z hz qz qz 2 qz 2

3.11

has a solution q ∈ Q1 . Let φ ∈ ΦS h, q , and f ∈ A with f zfz/z / 0. If zf z/fz ∈ Q1 and φ

zf z zf z 2 ,1 , z {f, z}; z fz f z

3.12

is univalent in U, then zf z 2 zf z ,1 , z {f, z}; z hz ≺ φ fz f z

3.13

12


implies qz ≺

zf z , fz

3.14

and q is the best subordinant. Proof. The proof is similar to the proof of Theorem 2.5, and is therefore omitted. Combining Theorems 2.3 and 3.3, we obtain the following sandwich-type theorem. Corollary 3.5. Let h1 and q1 be analytic functions in U, let h1 be an analytic univalent function in U, 0. q2 ∈ Q1 with q1 0 q2 0 1 and φ ∈ ΦS h2 , q2 ∩ ΦS h1 , q1 . Let f ∈ A with f zfz/z / 2 If zf z/fz ∈ H ∩ Q1 and φzf z/fz, 1 zf z/f z, z {f, z}; z is univalent in U, then

zf z 2 zf z ,1 , z {f, z}; z ≺ h2 z h1 z ≺ φ fz f z

3.15

implies q1 z ≺

zf z ≺ q2 z. fz

3.16

4. Schwarzian derivatives and convexity We introduce the following class of admissible functions. Definition 4.1. Let Ω be a set in C and q ∈ Q1 ∩ H. The class of admissible functions ΦSc Ω, q consists of those functions φ : C2 × U → C that satisfy the admissibility condition 1 − q2 ζ ∈ Ω, φ qζ, kζq ζ ;z / 2

4.1

z ∈ U, ζ ∈ ∂U \ Eq, and k ≥ 1. Theorem 4.2. Let φ ∈ ΦSc Ω, q , and f ∈ A with f z / 0. If zf z 2 , z {f, z}; z : z ∈ U ⊂ Ω, φ 1 f z

4.2

then 1

zf z ≺ qz. f z

4.3


13

Proof. Define the function p by pz : 1

zf z . f z

4.4

Clearly p ∈ A, and a simple calculation yields z2 {f, z} zp z

1 − p2 z . 2

4.5

1 − r2 . 2

4.6

Define the transformation from C2 to C2 by u r,

v s

Let 1 − r2 ψr, s; z φu, v; z φ r, s ;z . 2

4.7

The proof will make use of Lemma 1.3. Using 4.4 and 4.5, from 4.7, we obtain zf z 2 ψ pz, zp z; z φ 1 , z {f, z}; z . f z

4.8

ψ pz, zp z; z ∈ Ω.

4.9

Hence 4.2 becomes

From 4.7, we see that the admissibility condition for φ ∈ ΦSc Ω, q is equivalent to the admissibility condition for ψ as given in Definition 1.1. Hence ψ ∈ ΨΩ, q and by Lemma 1.3, pz ≺ qz or 1

zf z ≺ qz. f z

4.10

We will denote by ΦSc h, q the class ΦSc hU, q , where h is the conformal mapping of U onto Ω / C. Proceeding similarly as in the previous section, the following results can be established, which we state without proof. 0 satisfies Theorem 4.3. Let φ ∈ ΦSc h, q . If f ∈ A with f z /

zf z 2 , z {f, z}; z ≺ hz, φ 1 f z

4.11

14


then 1

zf z ≺ qz. f z

4.12

We extend Theorem 4.3 to the case where the behavior of q on ∂U is not known. Theorem 4.4. Let Ω ⊂ C and let q be univalent in U with q0 1. Let φ ∈ ΦSc h, qρ for some 0 satisfies 4.2, then 4.12 holds. ρ ∈ 0, 1 where qρ z qρz. If f ∈ A with f z / Theorem 4.5. Let Ω be a set in C, qz 1 Mz, M > 0, and φ : C2 × U → C satisfy 2k − 1 − Meiθ Meiθ ; z / ∈Ω φ 1 Meiθ , 2

4.13

whenever z ∈ U, θ ∈ R and k ≥ 1. Let f ∈ A with f z / 0. If

zf z 2 φ 1 , z {f, z}; z ∈ Ω, f z

4.14

zf z f z < M.

4.15

then

In the special case Ω qU {ω : |ω − 1| < M}, Theorem 4.5 gives the following: let φ : C2 × U → C satisfy iθ φ 1 Meiθ , 2k − 1 − Me Meiθ ; z − 1 ≥ M 2

4.16

whenever z ∈ U, θ ∈ R, and k ≥ 1; if f ∈ A with f z / 0 satisfies φ 1 zf z , z2 {f, z}; z − 1 < M, f z

4.17

zf z f z < M.

4.18

then

With φu, v; z u v, we get the following: Example 4.6. If 0 < M < 2, and f ∈ A with f z / 0 satisfies zf z 2 f z z {f, z} < M,

4.19


15

then zf z f z < M.

4.20

We next apply Theorem 4.2 to the particular case corresponding to qU being a halfplane qU Δ. Theorem 4.7. Let Ω be a set in C. Let φ : C2 × U → C satisfy the admissibility condition φiρ, η; z / ∈Ω

4.21

for all z ∈ U, and for all real ρ and η with η ≤ 0. Let f ∈ A with f z / 0. If zf z 2 , z {f, z}; z ∈ Ω, φ 1 f z

4.22

then f ∈ K. Let hz 1 z/1 − z. Clearly, hU Δ. Writing the class of admissible functions ΦSc hU, Δ as ΦSc Δ , the following result is a restatement of 1.6, which is an immediate consequence of Theorem 4.7. 0 Corollary 4.8 see 2, Theorem 4.6b, page 246 . Let φ ∈ ΦSc Δ . If f ∈ A with f z / satisfies zf z 2 , z {f, z}; z > 0, R φ 1 f z

4.23

then f ∈ K. Definition 4.9. Let Ω be a set in C and q ∈ H. The class of admissible functions ΦSc Ω, q consists of those functions φ : C2 × U → C that satisfy the admissibility condition zq z 1 − q2 z φ qz, ; ζ ∈ Ω, m 2

4.24

z ∈ U, ζ ∈ ∂U, and m ≥ 1. Now we will give the dual result of Theorem 4.2 for differential superordination. Theorem 4.10. Let φ ∈ ΦSc Ω, q , and f ∈ A with f z / 0. If 1 zf z/f z ∈ Q1 and 2 φ1 zf z/f z, z {f, z}; z is univalent in U, then zf z 2 , z {f, z}; z : z ∈ U Ω⊂ φ 1 f z

4.25

16


implies qz ≺ 1

zf z . f z

4.26

Proof. With pz 1 zf z/f z and 1 − r2 ; z φu, v; z, ψr, s; z φ r, s 2

4.27

from 4.8 and 4.25, we have

Ω ⊂ ψ pz, zp z; z : z ∈ U .

4.28

From 4.6, we see that the admissibility condition for φ ∈ ΦSc Ω, q is equivalent to the admissibility condition for ψ as given in Definition 1.2. Hence ψ ∈ Ψ Ω, q , and by Lemma 1.4, qz ≺ pz or qz ≺ 1

zf z . f z

4.29

Proceeding similarly as in the previous section, the following result is an immediate consequence of Theorem 4.10. 0. If Theorem 4.11. Let q ∈ H, let h be analytic in U and φ ∈ ΦSc h, q . Let f ∈ A with f z / 1 zf z/f z ∈ Q1 and φ1 zf z/f z, z2 {f, z}; z is univalent in U, then zf z 2 hz ≺ φ 1 , z {f, z}; z f z

4.30

implies qz ≺ 1

zf z . f z

4.31

Combining Theorems 4.3 and 4.11, we obtain the following sandwich-type theorem. Corollary 4.12. Let h1 and q1 be analytic functions in U, let h1 be analytic univalent in U, q2 ∈ Q1 with q1 0 q2 0 1 and φ ∈ ΦSc h2 , q2 ∩ ΦSc h1 , q1 . Let f ∈ A with f z / 0. If 2 1 zf z/f z ∈ H ∩ Q1 and φ 1 zf z/f z, z {f, z}; z is univalent in U, then zf z 2 h1 z ≺ φ 1 , z {f, z}; z ≺ h2 z f z

4.32


17

implies q1 z ≺ 1

zf z ≺ q2 z. f z

4.33

Acknowledgments The work presented here is supported by the FRGS and Science Fund research grants, and it was completed during V. Ravichandran visit to USM. The University’s support is gratefully acknowledged. References 1 S. Owa and M. Obradović, “An application of differential subordinations and some criteria for univalency,” Bulletin of the Australian Mathematical Society, vol. 41, no. 3, pp. 487–494, 1990. 2 S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Application, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000. 3 S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables. Theory and Application, vol. 48, no. 10, pp. 815–826, 2003. 4 R. M. Ali and V. Ravichandran, “Classes of meromorphic α-convex functions,” to appear in Taiwanese Journal of Mathematics. 5 R. M. Ali, V. Ravichandran, M. Hussain Khan, and K. G. Subramanian, “Differential sandwich theorems for certain analytic functions,” Far East Journal of Mathematical Sciences, vol. 15, no. 1, pp. 87–94, 2004. 6 R. M. Ali, V. Ravichandran, M. Hussain Khan, and K. G. Subramanian, “Applications of first order differential superordinations to certain linear operators,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 5, pp. 799–810, 2006. 7 R. M. Ali, V. Ravichandran, and S. K. Lee, “Subclasses of multivalent starlike and convex functions,” to appear in Bulletin of the Belgian Mathematical Society. Simon Stevin. 8 R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Differential subordination and superordination of analytic functions defined by the Dziok-Srivastava linear operator,” preprint. 9 R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Differential subordination and superordination of analytic functions defined by the multiplier transformation,” to appear in Mathematical Inequalities & Applications. 10 R. M. Ali, V. Ravichandran, and N. Seenivasagan, “Subordination and superordination of the LiuSrivastava linear operator on meromorphic functions,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 31, no. 2, pp. 193–207, 2008. 11 T. Bulboac˘a, “Sandwich-type theorems for a class of integral operators,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 13, no. 3, pp. 537–550, 2006. 12 T. Bulboac˘a, “A class of double subordination-preserving integral operators,” Pure Mathematics and Applications, vol. 15, no. 2-3, pp. 87–106, 2004. 13 T. Bulboac˘a, “A class of superordination-preserving integral operators,” Indagationes Mathematicae, vol. 13, no. 3, pp. 301–311, 2002. 14 T. Bulboac˘a, “Generalized Briot-Bouquet differential subordinations and superordinations,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 47, no. 5-6, pp. 605–620, 2002. 15 T. Bulboac˘a, “Classes of first-order differential superordinations,” Demonstratio Mathematica, vol. 35, no. 2, pp. 287–292, 2002. 16 N. E. Cho and S. Owa, “Double subordination-preserving properties for certain integral operators,” Journal of Inequalities and Applications, vol. 2007, Article ID 83073, 10 pages, 2007. 17 N. E. Cho and H. M. Srivastava, “A class of nonlinear integral operators preserving subordination and superordination,” Integral Transforms and Special Functions, vol. 18, no. 1-2, pp. 95–107, 2007. 18 S. S. Miller and P. T. Mocanu, “Briot-Bouquet differential superordinations and sandwich theorems,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 327–335, 2007.

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