Abstract. The purpose of the present paper is to introduce subclass of p−valent functions defined by certain linear operator and to investigate various subor- dination properties for this subclass.

SOME APPLICATIONS OF SUBORDINATION FOR HIGHER-ORDER DERIVATIVES OF p−VALENT FUNCTIONS

DEFINED BY CERTAIN LINEAR OPERATOR

T. M. Seoudy

(Received 27 July, 2012)

Abstract. The purpose of the present paper is to introduce subclass of p−valent functions defined by certain linear operator and to investigate various subor- dination properties for this subclass.

1. Introduction

Let A (p, k) denote the class of functions of the form:

f (z) = z ^p +

∞

X

n=p+k

a n z ⁿ (p, k ∈ N = {1, 2, 3, ...}) (1.1)

which are analytic in the open unit disk U = {z ∈ C : |z| < 1} . For simplicity, we write A (p, 1) = A (p) and A (1, 1) = A. If f (z) and g (z) are analytic in U , we say that f (z) is subordinate to g (z) or g (z) is superordinate to f (z) , written as f ≺ g in U or f (z) ≺ g(z) (z ∈ U ), if there exists a Schwarz function ω (z), which (by definition) is analytic in U with ω (0) = 0 and |ω (z)| < 1 (z ∈ U ) such that f (z) = g(ω(z)) (z ∈ U ). Furthermore, if the function g (z) is univalent in U , then we have the following equivalence holds (see [9] and [10]) :

f (z) ≺ g(z) ⇐⇒ f (0) = g(0) and f (U ) ⊂ g(U ).

Upon differentiating both sides of (1.1) j−times with respect and to z, we have

f ^(j) (z) = δ (p; j) z ^p−j +

∞

X

n=p+k

δ (n; j) a n z ^n−j , (1.2)

where

δ (p; j) = p!

(p − j)! (p > j; p ∈ N; j ∈ N 0 = N ∪ {0}) . (1.3)

2010 Mathematics Subject Classification 30C45.

Key words and phrases: Analytic function, Differential subordination, p−valent function, Linear

operator.

For a function f ^(j) (z) given by (1.2) , Aouf and Seoudy [4, 5] defined the linear operator D _p ⁿ f ^(j) by:

D _p ⁰ f ^(j) (z) = f ^(j) (z) , D _p ¹ f ^(j) (z) = D 

f ^(j) (z) 

= δ (p; j) z ^p−j +

∞

X

n=p+k

δ (n; j)  n − j p − j



a n z ^n−j ,

D _p ² f ^(j) (z) = D 

D _p ¹ f ^(j) (z) 

= δ (p; j) z ^p−j +

∞

X

n=p+k

δ (n; j)  n − j p − j

 ²

a _nk z ^n−j ,

and (in general)

D _p ^m f ^(j) (z) = D(D _p ^m−1 f ^(j) (z)) = δ (p; j) z ^p−j +

∞

X

n=p+k

δ (n; j)  n − j p − j

 m

a _n z ^n−j (p > j; p, m ∈ N; j ∈ N ⁰ ; z ∈ U ) . (1.4) From (1.4), we can easily deduce that

z 

D ^m _p f ^(j) (z) 

= (p − j) D ^m+1 _p f ^(j) (z) (p > j; p ∈ N; m, j ∈ N 0 ; z ∈ U ) . (1.5) The operator D _p ^m f ^(j) (z) (p > j, p ∈ N, m, j ∈ N ⁰ ) was introduced and studied by Aouf [1, 2] where

f (z) = z ^p −

∞

X

n=p+1

a n z ⁿ (a n ≥ 0) . We note that:

(i) D _p ⁿ f ⁽⁰⁾ (z) = D ^m _p f (z) was introduced and studied by Kamali and Orhan [8]

and Aouf and Mostafa [3];

(ii) D ^m ₁ f ⁽⁰⁾ (z) = D ^m f (z) was introduced by S˘ al˘ agean [11].

By making use of the linear operator D ^m _p f ^(j) (z) and the principle of subor- dination between analytic functions, we now introduce the following subclass of p−valent functions.

For fixed parameters A, B (−1 ≤ B < A ≤ 1) and α (0 ≤ α < δ (p; j)), we say that a function f ∈ A (p, k) is in the class C _p ^j (m; A, B, α), if it satisfies the following subordination condition :

D ^m _p f ^(j) (z)

z ^p−j ≺ δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z

1 + Bz . (1.6)

In view of the definition of subordination, (1.6) is equivalent to the following con- dition :

D

f

(z)

z

− δ (p; j)

B ^D

_z ^f

^(z) − [δ (p; j) B + (A − B) (δ (p; j) − α)]

< 1 (z ∈ U ) .

For convenience, we write

C ^j _p (m; 1, −1, α) = C _p ^j (m; α) ,

where C _p ^j (m; α) denotes the class of functions f (z) ∈ A (p, k) satisfying the following inequality :

Re

( D _p ^m f ^(j) (z) z ^p−j

)

> α (0 ≤ α < δ (p; j) ; z ∈ U ) .

In the present paper, we aim at proving such results as subordination and super- ordination properties, convolution properties, distortion theorems and inequality properties of the class C _p ^j,µ (m; A, B, λ).

2. Main Results

In order to establish our main results, we need the following lemmas.

Lemma 1 [7, 10] . Let the function h (z) be analytic and convex (univalent) in U with h (0) = c. Suppose also that the function g (z) given by

g (z) = c + c k z ^k + c k+1 z ^k+1 + ... (2.1) is analytic in U. If

g (z) + zg

(z)

γ ≺ h (z) (< (γ) > 0; γ 6= 0) , (2.2) then

g (z) ≺ q (z) = γ k z ⁻

Z

h (t) t

⁻¹ dt ≺ h (z) , and q (z) is the best dominant of (2.2).

For real or complex numbers a, b and c (c / ∈ Z ⁻ 0 ), the Gaussian hypergeometric function is defined by

2 F 1 (a, b; c; z) = 1 + ab c . z

1! + a(a + 1)b(b + 1) c(c + 1) . z ²

z! + ... . (2.3) We note that the above series converges absolutely for z ∈ U and hence represents an analytic function in U (see, for details [12, Chapter 14]).

Each of the identities (asserted by Lemma 4 below) is well-known (cf., e.g., [12, Chapter 14]).

Lemma 2 [12]. For real or complex parameters a, b and c (c / ∈ Z ⁻ 0 ),

1

Z

0

t ^b−1 (1 − t) ^c−b−1 (1 − zt) ^−a dt = Γ(b)Γ(c − b)

Γ(c) ² Γ 1 (a, b; c; z) (Re(c) > Re(b) > 0) ; (2.4)

2 F ₁ (a, b; c; z) = (1 − z) ^−a ₂ F ₁ (a, b; c; z

z − 1 ) ; (2.5)

2 F ₁ (a, b; c; z) = ₂ F ₁ (a, b − 1; c; z) + az

c ² F ₁ (a + 1, b; c + 1; z) ; (2.6)

2 F 1 (a, b; a + b + 1 2 ; 1

2 ) =

√ πΓ( ^a+b+1 ₂ )

Γ( ^a+1 ₂ )Γ( ^b+1 ₂ ) . (2.7) Unless otherwise mentioned, we assume throughout this paper that µ > 0, p >

j, p ∈ N and m, j ∈ N 0 .

Theorem 1. Let the function f defined by (1.1) satisfying the following subordi- nation condition :

(1 − µ)D _p ^m f ^(j) (z) + µD ^m+1 _p f ^(j) (z)

z ^p−j ≺ δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z

1 + Bz .

Then

D ^m _p f ^(j) (z)

z ^p−j ≺ Q(z) ≺ δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z

1 + Bz , (2.8)

where the function Q given by Q(z) =

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1+Bz) 2 F 1



1, 1; ^p−j _µ + 1; _1+Bz ^Bz 

(B 6= 0) 1 + (p−j)A(δ(p;j)−α)

p−j+µ z (B = 0)

is the best dominant. Furthermore, Re

( D ^m _p f ^(j) (z) z ^p−j

)

> ξ (z ∈ U ) , (2.9)

where ξ =

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1−B) 2 F ₁ 

1, 1; ^p−j _µ + 1; _B−1 ^B 

(B 6= 0) 1 − (p−j)A(δ(p;j)−α)

p−j+µ z (B = 0)

The estimate in (2.9) is the best possible.

Proof. Consider the function g defined by g(z) = D ^m _p f ^(j) (z)

z ^p−j (z ∈ U ) . (2.10)

Then g is of the form (2.1) with c = δ (p; j) and is analytic in U . Differentiating (2.10) with respect to z and using (1.5), we obtain

(1 − µ)D _p ^m f ^(j) + µD _p ^m+1 f ^(j) (z)

z ^p−j = g(z) + µ

p − j zg

(z) ≺ 1 + Az 1 + Bz . Now, by using Lemma 1 for γ = ^p−j _µ , we obtain

D ^m _p f ^(j)

z ^p−j ≺ G(z) = p − j µ z ⁻

z

Z

0

t

⁻¹  δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] t 1 + Bt

 dt

=

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1+Bz) 2 F 1



1, 1; ^p−j _µ + 1; _1+Bz ^Bz 

(B 6= 0) 1 + (p−j)A(δ(p;j)−α)

p−j+µ z (B = 0)

by change of variables followed by the use of the identities (2.4), (2.5) and (2.6) (with a = 1, c = b + 1, b = ^p−j _µ ). This proves the assertion (2.8) of Theorem 1.

Next, in order to prove the assertion (2.9) of Theorem 1, it suffices to show that inf

|z|<1 {Re(G(z))} = G(−1) . (2.11) Indeed we have, for |z| ≤ r < 1,

Re  δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z 1 + Bz



≥ δ (p; j) − [δ (p; j) B + (A − B) (δ (p; j) − α)] r

1 − Br .

Upon setting

G(s, z) = δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] sz 1 + Bsz

and

dν(s) = p − j

µ s

⁻¹ ds (0 ≤ s ≤ 1) , which is a positive measure on the closed interval [0, 1], we get

Q(z) =

1

Z

0

G(s, z)dν(s) ,

so that Re {Q(z)} ≥

1

Z

0

 δ(p;j)−[δ(p;j)B+(A−B)(δ(p;j)−α)]r 1−Br

 dν(s) = Q(−r) (|z| ≤ r < 1) .

Letting r → 1 ⁻ in the above inequality, we obtain the assertion (2.9) of Theorem 1.

Finally, the estimate in (2.9) is the best possible as the function Q is the best dominant.

Taking µ = 1 in Theorem 1, we obtain the following corollary.

Corollary 1. The following inclusion property holds true for the class C _p ^j (m; A, B, α):

C _p ^j (m + 1; A, B, α) ⊂ C _p ^j (m; β) ⊂ C _p ^j (m; A, B, α) , where

β =

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1−B) 2 F 1



1, 1; p − j + 1; _B−1 ^B 

(B 6= 0) 1 − (p−j)A(δ(p;j)−α)

p−j+1 z (B = 0)

The result is the best possible.

Taking µ = 1, A = 1 and B = −1 in Theorem 1, we obtain the following corollary.

Corollary 2. The following inclusion property holds true for the function class C _p ^j (m + 1; α) ⊂ C _p ^j (m; β) ⊂ C _p ^j (m; α) ,

where

β = α + (δ (p; j) − α)



2 F 1



1, 1; p − j + 1; 1 2



− 1

 . The result is the best possible.

Theorem 2. Let f ∈ C _p ^j (m; A, B, α) and let

F _c ^(j) (f )(z) = c + p − j z ^c

z

Z

0

t ^c−1 f ^(j) (t)dt (c > 0; z ∈ U ) . (2.12)

Then F c ^(j) (f ) ∈ C _p ^j (m; A, B, α) and D ^m _p F c ^(j) (f )(z)

z ^p−j ≺ Q(z) ≺ δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z

1 + Bz , (2.13)

where the function Q given by Q(z) =

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1+Bz) 2 F ₁ 

1, 1; c + p − j + 1; _1+Bz ^Bz 

(B 6= 0) 1 + (c+p−j)A(δ(p;j)−α)

c+p−j+1 z (B = 0)

is the best dominant. Furthermore, Re

( D _p ^m F c ^(j) (f )(z) z ^p−j

)

> ξ (z ∈ U ) , (2.14)

where ξ =

( δ(p;j)B+(A−B)(δ(p;j)−α)

B + (B−A)(δ(p;j)−α) B(1−B) 2 F 1



1, 1; c + p − j + 1; _B−1 ^B 

(B 6= 0) 1 − (c+p−j)A(δ(p;j)−α)

c+p−j+1 (B = 0).

The result is the best possible.

Proof. Defining the function g by

g(z) = D ^m _p F c ^(j) (f )(z)

z ^p−j (z ∈ U ) , (2.15)

we note that g is of the form (2.1) with c = δ (p; j) and is analytic in U . Differen- tiating (2.15) with respect to z and using the following operator identity :

z(D _p ^m F _c ^(j) (f )(z))

= (c + p − j) D ^m _p F _c ^(j) (f )(z) − cD ^m _p F _c ^(j) (f )(z) (2.16) in the resulting equation, we find that

D ^m _p f ^(j) (z)

z ^p−j = g(z) + zg

(z)

c + p − j ≺ δ (p; j) + [δ (p; j) B + (A − B) (δ (p; j) − α)] z

1 + Bz .

Now the remaining part of Theorem 2 follows by employing the techniques that we used in proving Theorem 1 above.

Theorem 3. If f ∈ C _p ^j (m; α) (0 ≤ α < δ (p; j)), then

Re

( (1 − µ)D ^m _p f ^(j) (z) + µD _p ^m+1 f ^(j) (z) z ^p−j

)

> α (|z| < R) , (2.17)

where

R = (s

1 + µ ²

(p − j) ² − µ p − j

)

. (2.18)

The result is the best possible.

Proof. Since f ∈ C _p ^j (m; α) , we write D ^m _p f ^(j) (z)

z ^p−j = α + (δ (p, j) − α)u(z) (z ∈ U ) . (2.19) Then, clearly, u is of the form (2.1) with c = 1, is analytic in U , and has a positive real part in U . Differentiating (2.19) with respect to z and using (1.5), we obtain

1 δ (p; j) − α

( (1 − µ)D _p ^m f ^(j) (z) + µD ^m+1 _p f ^(j) (z)

z ^p−j − α

)

= u(z) + µ

p − j zu

(z) . (2.20) Now, by applying the well-known estimate [6]

   zu

(z) 

Re{u(z)} ≤ 2kr ^k

1 − r ^2k (|z| = r < 1) in (2.20), we obtain

Re



 



 



(1 − µ)D _p ^m f ^(j) (z) + µD _p ^m+1 f ^(j) (z)

z ^p−j − α

δ (p; j) − α



 



 



≥ Re{u(z)} .



1 − 2µkr ^k (p − j) (1 − r ^2k )

 .

(2.21) It is easily seen that the right-hand side of (2.21) is positive provided that r < R, where R is given as in Theorem 3. This proves the assertion (2.17) of Theorem 3.

In order to show that the bound R is the best possible, we consider the function f ∈ A (p, k) defined by

D _p ^m f ^(j) (z)

z ^p−j = α + (δ (p, j) − α) 1 + z ^k

1 − z ^k (z ∈ U ) . Noting that

1 δ (p; j) − α

( (1 − µ)D ^m _p f ^(j) (z) + µD _p ^m+1 f ^(j) (z)

z ^p−j − α

)

= p − j − (p − j) z ^2k + 2µkz ^k (p − j) (1 − z ^k ) ² = 0 for z = R

exp ^iπ _k
, we complete the proof of Theorem 3.

Putting µ = 1 in Theorem 3, we obtain the following result.

Corollary 3. If f ∈ C _p ^j (m; α) (0 ≤ α < δ (p; j)), then f ∈ C _p ^j (m + 1; α) for

|z| < R ^∗ ,where

R ^∗ = (s

1 + 1

(p − j) ² − 1 p − j

)

.

The result is the best possible.

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T. M. Seoudy

Department of Mathematics, Faculty of Science,

Fayoum University, Fayoum 63514, Egypt

[email protected]