Properties of certain subclasses of meromorphic functions with positive coefficients

Mathematical and Computer Modelling 49 (2009) 868–879

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Properties of certain subclasses of meromorphic functions with positive coefficients M.K. Aouf, R.M. El-Ashwah ∗ Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

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Article history: Received 15 May 2007 Received in revised form 15 March 2008 Accepted 25 April 2008 Keywords: Linear operator Meromorphic functions Neighborhoods Hadamard product

In this paper, we obtain coefficient estimates, distortion theorem, radii of starlikeness ∗ and convexity for the class Σm (a, c , A, B, α) of meromorphic functions with positive ∗ coefficients. Some properties of neighborhoods of functions in the class Σm (a, c , A, B, α) are investigated. Also we derive many results for the Hadamard products of functions ∗ belonging to the class Σm (a, c , A, B, α). © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Let Σp denote the class of meromorphic functions of the form: f (z ) =

1 z

+

∞ X

an z n

(p ∈ N ∗ = {2i − 1 : i ∈ N = 1, 2, . . .}),

(1.1)

n =p

which are analytic in the punctured disc U ∗ = {z : 0 < |z | < 1} = U \ {0} with a simple pole at the origin with residue one there. A function f ∈ Σp is said to be meromorphically starlike of order α if it satisfies

( Re −

0

zf (z ) f (z )

) >α

(1.2)

for some α (0 ≤ α < 1) and for all z ∈ U ∗ . Further, a function f ∈ Σp is said to be meromorphically convex of order α if it satisfies

( Re − 1 +

00

zf (z ) f (z ) 0

!) >α

(1.3)

for some α(0 ≤ α < 1) and for all z ∈ U ∗ . Some subclasses of Σp when p = 1 were introduced and studied by Pommerenke [1], Miller [2], Mogra et al. [3], Cho [4], Cho et al. [5] and Aouf [6,7].

∗

Corresponding author. E-mail addresses: [email protected] (M.K. Aouf), [email protected] (R.M. El-Ashwah).

0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.04.013

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

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For functions f ∈ Σp given by (1.1) and g ∈ Σp given by g (z ) =

1

+

z

∞ X

bn z n

(p ∈ N ∗ ),

(1.4)

n =p

we define the Hadamard product (or convolution) of f and g by

(f ∗ g )(z ) =

1 z

+

∞ X

an bn z n .

(1.5)

n =p

In terms of the Pochhammer symbol (λ)n given by

 Γ (λ + n) 1 (λ)n = = λ(λ + 1) · · · (λ + n − 1) Γ (λ) we now define the function ϕ(a, c ; z ) by ϕ(a, c ; z ) =

1 z

+

∞ X (a)n n=1

(c )n

(n = 0) (n ∈ N = {1, 2, 3, . . .}),

(1.6)

(z ∈ U ∗ ; a ∈ R; c ∈ R \ Z0− ; Z0− = {0, −1, −2, . . .}).

z n−1

(1.7)

Corresponding to the function ϕ(a, c ; z ), we define the linear operator L(a, c ) (see Liu [8] and Liu and Srivastava [9]), which is defined by means of the following Hadamard product (or convolution): L(a, c )f (z ) = ϕ(a, c ; z ) ∗ f (z )

(f ∈ Σp ).

(1.8)

Just as in [8] and [9], it is easily verified from the definitions (1.7) and (1.8) that 0

z (L(a, c )f (z )) = aL(a + 1, c )f (z ) − (a + 1)L(a, c )f (z ).

(1.9)

Moreover, we observe that, for f ∈ Σp , 0

L(a, a)f (z ) = f (z ) and L(2, 1)f (z ) = 2f (z ) + zf (z ). Let Σp∗ be the subclass of Σp consisting of functions of the form: f (z ) =

1 z

+

∞ X

|an | z n (p ∈ N ∗ ).

(1.10)

n=p

Making use of the operator L(a, c ), we now introduce a subclass of Σp∗ as defined below. For fixed parameters A, B and α(−1 ≤ A < B ≤ 1; 0 < B ≤ 1; 0 ≤ α < m!; m ∈ N ∗ ; m ≤ p) we say that a function ∗ f ∈ Σp∗ is in the class Σm (a, c , A, B, α) if it also satisfies the following inequality:

z m+1 (L(a, c )f (z ))(m) + m! Bz m+1 (L(a, c )f (z ))(m) + [m!B + (A − B)(m! − α)] < 1 (z ∈ U ).

(1.11)

We note that (i) Σp∗ (a, a, −β, β, 0) = Tp (β) (p ∈ N ∗ and 0 < β ≤ 1) (Kim et al. [10]);

(ii) Σ1∗ (a, a, −β, (2γ − 1)β, α) = Σ1 (α, β, γ ) (0 ≤ α < 1; 0 < β ≤ 1 and 21 ≤ γ ≤ 1) (Cho et al. [5]); (iii) Σ1∗ (a, a, −A, −B, 0)P = Σp (A, B) (−1 ≤ B < A ≤ 1 and −1 ≤ B < 0) (Cho [4]); (α, 1, A, B, 1) (0 ≤ α < 1; −1 ≤ A < B ≤ 1 and 0 < B ≤ 1) (Aouf [7]). (iv) Σ1∗ (a, a, A, B, α) = Also we observe that: (i)

Σp∗ (a, a, −β, β, α) = Σp∗ (α, β)  ∗ = f ∈ Σp :

z p+1 f (p) (z ) + p!

 < β (z ∈ U ; 0 ≤ α < p!; p ∈ N ∗ ; 0 < β ≤ 1) ; z p+1 f (p) (z ) − p! + 2α (1.12)

(ii)

Σp∗ (a, a, −β, (2γ − 1)β, α) = Σp∗ (α, β, γ )  z p+1 f (p) (z ) + p! < β, = f ∈ Σp∗ : p + 1 ( p ) (2γ − 1)z f (z ) + (2γ α − p!)

(z ∈ U ; 0 ≤ α < p!; p ∈ N ; ∗

1 2



≤ γ ≤ 1; 0 < β ≤ 1) .

(1.13)

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M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

(iii)

Σ1∗ (a, c , A, B, α) = Σ ∗ (a, c , A, B, α)  1 + [B + (A − B)(1 − α)]z 0 = f ∈ Σp∗ : z 2 (L(a, c )f (z )) ≺ − −, 1 + Bz  (z ∈ U ; −1 ≤ A < B ≤ 1; 0 < B ≤ 1; 0 ≤ α < 1) .

(1.14)

2. Coefficient estimates ∗ Theorem 1. Let f ∈ Σp∗ be given by (1.10). Then f ∈ Σm (a, c , A, B, α) if and only if ∞ X

(a)n+1 n |an | ≤ (B − A)(m! − α), (1 + B) m (c )n+1

(2.1)

n! . (n − m)!m!

(2.2)

  m!

n=p

where

  n m

=

∗ Proof. Let f ∈ Σm (a, c , A, B, α) is given by (1.10). Then, from (1.10) and (1.11), we have

∞ P (a)n+1 n! n +1 | | a z n (n−m)! (c )n+1 m+1 (m) z (L(a, c )f (z )) + m! n=p = ∞ Bz m+1 (L(a, c )f (z ))(m) + [m!B + (A − B)(m! − α)] (B − A)(m! − α) − B P n! (a)n+1 |a | z n+1 n (n−m)! (c )n+1 n =p

< 1 (z ∈ U ).

(2.3)

Since |Re(z )| ≤ |z | for any z, choosing z to be real and letting z → 1 through real values, (2.3) yields −

∞ X n=p

∞ X (a)n+1 n! (a)n+1 |an | ≤ (B − A)(m! − α) − B |an | , (n − m)! (c )n+1 ( n − m )! (c )n+1 n =p

n!

(2.4)

which leads us at once to (2.1). In order to prove the converse, we assume that the inequality (2.1) holds true. Then, if we let z ∈ ∂ U, we find from (1.10) and (2.1) that

z m+1 (L(a, c )f (z ))(m) + m! Bz m+1 (L(a, c )f (z ))(m) + [m!B + (A − B)(m! − α)] ≤

∞ P n =p

(a)n+1 n! (n−m)! (c )n+1

(B − A)(m! − α) − B

∞ P n =p

< 1 (z ∈ ∂ U ). ∗ Hence, by the maximum modulus theorem, we have f ∈ Σm (a, c , A, B, α).

|an |

(a)n+1 n! (n−m)! (c )n+1

| an | (2.5)



∗ Corollary 1. Let f ∈ Σp∗ be given by (1.10). If f ∈ Σm (a, c , A, B, α), then

|an | ≤

(B − A)(m! − α)(c )n+1   n m! (1 + B)(a)n+1

(n ≥ p ≥ m; p, m ∈ N ∗ ).

(2.6)

m

The result is sharp for the function f given by f (z ) =

1 z

+

(B − A)(m! − α)(c )n+1 n   z (n ≥ p ≥ m; p, m ∈ N ∗ ). n m! (1 + B)(a)n+1 m

(2.7)

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

871

3. Distortion theorem ∗ Theorem 2. If a function f defined by (1.10) is in the class Σm (a, c , A, B, α), then

 (c )p+1 (B − A)(m! − α)(p − m)! p+1 −(j+1) (j) r r ≤ f (z ) (a)p+1 (1 + B)(p − j)!    (c )p+1 (B − A)(m! − α)(p − m)! p+1 −(j+1) ≤ j! + r r 0 < |z | = r < 1; a > c > 0; (a)p+1 (1 + B)(p − j)!  (a)p+1 (1 + B)j!(p − j)! m! − ≤ α < m!; p, m ∈ N ∗ ; j ∈ N0 = {0, 1, 2, . . .}; 0 ≤ j ≤ m ≤ p . (c )p+1 (B − A)(p − m)!



j! −

(3.1)

The result is sharp for the functions f given by f (z ) =

1 z

+

(B − A)(m! − α)(c )p+1 p   z (p, m ∈ N ∗ , p ≥ m). p m! (1 + B)(a)p+1

(3.2)

m

Proof. In view of Theorem 1, we have ∞ ∞ X (p − j)!(1 + B)(a)p+1 X n! (a)n+1 n! |an | ≤ | an | (1 + B) (p − m)!(c )p+1 ( n − j )! ( n − m )! (c )n+1 n=p n =p

≤ (B − A)(m! − α), which yields ∞ X n =p

n! (c )p+1 (B − A)(m! − α)(p − m)! |an | ≤ (p, m ∈ N ∗ ; 0 ≤ j ≤ m ≤ p). (n − j)! (a)p+1 (1 + B)(p − j)!

(3.3)

Now, by differentiating both sides of (1.10) j times with respect to z, we have f (j) (z ) =

(−1)(j) j! z j +1

+

∞ X

n!

n =p

(n − j)!

|an | z n−j (j ∈ N0 ; p, m ∈ N ∗ ; 0 ≤ j ≤ m ≤ p),

and Theorem 2 follows easily from (3.3) and (3.4). Finally, it is easy to see that the bounds in (3.1) are attained for the function f given by (3.2).

(3.4)



Putting j = 0 in Theorem 2, we have ∗ Corollary 2. If a function f defined by (1.10) is in the class Σm (a, c , A, B, α), then

1

(c )p+1 (B − A)(m! − α)(p − m)! p 1 (c )p+1 (B − A)(m! − α)(p − m)! p r ≤ |f (z )| ≤ + r (a)p+1 (1 + B)p! r (a)p+1 (1 + B)p!  (a)p+1 (1 + B)p! ∗ 0 < |z | = r < 1; a > c > 0; m! − ≤ α < m!; p, m ∈ N ; p ≥ m . (c )p+1 (B − A)(p − m)!

− r 

(3.5)

Equalities in (3.5) are attained for the function f given by (3.2). Putting j = 1 in Theorem 2, we have ∗ Corollary 3. If a function f defined by (1.10) is in the class Σm (a, c , A, B, α), then

1

(c )p+1 (B − A)(m! − α)(p − m)! p−1 0 1 (c )p+1 (B − A)(m! − α)(p − m)! p−1 r ≤ f (z ) ≤ 2 + r (a)p+1 (1 + B)(p − 1)! r (a)p+1 (1 + B)(p − 1)!  (a)p+1 (1 + B)(p − 1)! 0 < |z | = r < 1; a > c > 0; m! − ≤ α < m!; p, m ∈ N ∗ ; p ≥ m . (c )p+1 (B − A)(p − m)!

− r2 

Equalities in (3.6) are attained for the function f given by (3.2).

(3.6)

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M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

4. Radii of starlikeness and convexity ∗ Theorem 3. Let the function f defined by (1.10) be in the class Σm (a, c , A, B, α). Then we have (i) f is meromorphically starlike of order δ (0 ≤ δ < 1) in the disc |z | < r1 , that is,

(

0

Re −

zf (z )

) > δ (|z | < r1 ; 0 ≤ δ < 1),

f (z )

(4.1)

where

r1 = inf

n ≥p

   

 n+1 1   

 

n m! (1 + B)(a)n+1 (1 − δ) m

 (B − A)(m! − α)(c )n+1 (n + 2 − δ)     

.

(4.2)

(ii) f is meromorphically convex of order δ(0 ≤ δ < 1) in the disc |z | < r2 , that is, ( !) 00 zf (z ) Re − 1 + 0 > δ (|z | < r2 ; 0 ≤ δ < 1), f (z )

(4.3)

where

r2 = inf

n ≥p

 

   

n m! (1 + B)(a)n+1 (1 − δ) m

 n+1 1   

 (B − A)(m! − α)(c )n+1 n(n + 2 − δ)     

.

(4.4)

Each of these results is sharp for the function f given by (2.7). Proof. (i) From the definition (1.10), we easily get

0 zf (z ) + 1 ≤ f (z )

∞ P

(n + 1) |an | |z |n+1

n =p

1−

∞ P

.

(4.5)

|an | |z |n+1

n =p

Thus, we have the desired inequality

0 zf (z ) + 1 ≤ 1 − δ (0 ≤ δ < 1), f (z )

(4.6)

if ∞ X (n + 2 − δ) n=p

(1 − δ)

|an | |z |n+1 ≤ 1.

(4.7)

Hence, by Theorem 1, (4.7) will be true if

  n

m! (1 + B)(a)n+1 m (n + 2 − δ) n+1 |z | ≤ (1 − δ) (B − A)(m! − α)(c )n+1

(n ≥ p ≥ m)

(4.8)

or

|z | ≤

   

  m!

n (1 + B)(a)n+1 (1 − δ) m

 n+1 1   

 (B − A)(m! − α)(c )n+1 (n + 2 − δ)     

(n ≥ p ≥ m).

(4.9)

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

873

Setting |z | = r1 in (4.9), the result follows. (ii) In order to prove the second assertion of Theorem 3, we find from the definition (1.10) that ∞ P

00 zf (z ) + 2 ≤ 0 f (z )

n(n + 1) |an | |z |n+1

n =p

1−

∞ P

.

(4.10)

n |an | |z |n+1

n=p

Thus we have the desired inequality

00 zf (z ) + 2 ≤ 1 − δ (0 ≤ δ < 1), 0 f (z )

(4.11)

if ∞ X n(n + 2 − δ) n =p

(1 − δ)

|an | |z |n+1 ≤ 1.

(4.12)

Hence, by Theorem 1, (4.12) will be true if

  n(n + 2 − δ)

(1 − δ)

m! n+1

|z |

≤

n (1 + B)(a)n+1 m

(B − A)(m! − α)(c )n+1

(n ≥ p ≥ m)

or

|z | ≤

 n+1 1   

 

   

n m! (1 + B)(a)n+1 (1 − δ) m

 (B − A)(m! − α)(c )n+1 n(n + 2 − δ)     

Setting |z | = r2 in (4.13), the result follows.

(n ≥ p ≥ m).

(4.13)



5. Neighborhoods Following the earlier works (based upon the familiar concept of neighborhoods of analytic functions) by Goodman [11] and Ruscheweyh [12], and (more recently) by Altintas et al. [13–15], Liu [8] and Liu and Srivastava [9], we begin by introducing here the δ -neighborhood of a function f ∈ Σp of the form (1.1) by means of the definition given below:

Nδ (f ) =

   

  g : g ∈ Σp , g ( z ) =

  

1 z

+

∞ X n =p

n

bn z and

∞ X n =p

m!

n (1 + |B|) m

(B − A)(m! − α)

·

(a)n+1 |bn − an | ≤ δ, (c )n+1

    ∗ (a > 0; c > 0; −1 ≤ A < B ≤ 1; δ > 0; 0 ≤ α < m!; p, m ∈ N ; p ≥ m) .   

(5.1)

Making use of the definition (5.1), we now prove Theorem 4 below: Theorem 4. Let the function f defined by (1.1) be in the class Σm (a, c , A, B, α). If f satisfies the following condition: f (z ) +  z −1 1+

∈ Σm (a, c , A, B, α) ( ∈ C , || < δ, δ > 0),

(5.2)

then Nδ (f ) ⊂ Σm (a, c , A, B, α).

(5.3)

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M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

Proof. It is easily seen from (1.11) that g ∈ Σm (a, c , A, B, α) if and only if for any complex number σ with |σ | = 1, z m+1 (L(a, c )g (z ))(m) + m!

(L(a, c )g (z ))(m) + [m!B + (A − B)(m! − α)]

z m+1 B

6= σ (z ∈ U ),

(5.4)

which is equivalent to

(g ∗ h)(z )

6= 0 (z ∈ U ),

z −1

(5.5)

where, for convenience, h(z ) =

=

1 z

∞ X

cn z n

n=p

1 z

+

+

    n    ∞  m! m (1 − σ B)   (a) X

n +1

n=p

 (B − A)(m! − α)σ  (c )    n+1 

zn.

(5.6)

From (5.6), we have

     n    m ! ( 1 − σ B )  (a)  m n + 1 |cn | = (B − A)(m! − α)σ  (c )n+1        m!

n (1 + |B|)(a)n+1 m

(n ≥ p ≥ m; p, m ∈ N ∗ ). (B − A)(m! − α) (c )n+1 P∞ Now, if f (z ) = 1z + n=p an z n ∈ Σp satisfies the condition (5.2), then (5.5) yields (f ∗ h)(z ) ≥ δ (z ∈ U ; δ > 0). −1 ≤

(5.7)

(5.8)

z

By Letting g (z ) =

1 z

+

∞ X

bn z n ∈ Nδ (f ),

(5.9)

n=p

so that

∞ X [g (z ) − f (z )] ∗ h(z ) = (bn − an )cn z n+1 n =p z −1   n ∞ m! m (1 + |B|) X (a)n+1 |bn − an | ≤ |z |p+1 ( B − A )( m ! − α) (c )n+1 n =p < δ (z ∈ U ; δ > 0).

(5.10)

Thus we have (5.5), and hence also (5.4) for any σ ∈ C such that |σ | = 1, which implies that g ∈ Σm (a, c , A, B, α). This evidently proves the inclusion (5.3) of Theorem 4. We now define the δ -neighborhood of a function f ∈ Σp∗ of the form (1.10) as follows:

   

  n

∞ m! m (1 + B) X (a)n+1 |bn | z n and ||bn | − |an || ≤ δ, Nδ+ (f ) = g ∈ Σp∗ : g (z ) = +  z (B − A)(m! − α) (c )n+1  n=p n =p 

1

∞ X

   

a > 0; c > 0; −1 ≤ A < B ≤ 1; 0 < B ≤ 1; δ > 0; 0 ≤ α < m!; p, m ∈ N ∗ ; p ≥ m

  

. 

(5.11)

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

875

∗ Theorem 5. . Let the function f defined by (1.10) be in the class Σm (a + 1, c , A, B, α), −1 ≤ A < B ≤ 1, 0 < B ≤ 1, 0 ≤ α < m!, p, m ∈ N ∗ and p ≥ m. Then

Nδ+ (f )

⊂ Σm (a, c , A, B, α) ∗



δ=



p+1 a+p+1

.

(5.12)

The result is sharp. Proof. Making use of the same method as in the proof Theorem 4, we can show that [cf. Eq. (5.6)] h( z ) =

1 z

+

∞ X

cn z n

n =p

  n

=

1 z

+

m! (1 − σ B)(a)n+1 ∞ X m n =p

(B − A)(m! − α)σ (c )n+1

zn.

(5.13)

∗ Thus, under the hypothesis −1 ≤ A < B ≤ 1, 0 < B ≤ 1, 0 ≤ α < m!, p, m ∈ N ∗ and p ≥ m, if f ∈ Σm (a + 1, c , A, B, α) is given by (1.10), we obtain

∞ X (f ∗ h)(z ) n + 1 cn |an | z z −1 = 1 + n=p   n ∞ m! m (1 + B) X (a)n+1 | an | ≥ 1− (B − A)(m! − α) (c )n+1 n =p   n ∞ m! m (1 + B) X a (a + 1)n+1 | an | . ≥ 1− a + p + 1 n=p (B − A)(m! − α) (c )n+1 Also, from Theorem 1, we obtain

(f ∗ h)(z ) a p+1 z −1 ≥ 1 − a + p + 1 = a + p + 1 = δ. The remaining part of the proof of Theorem 5 is similar to that of Theorem 4, and we skip the details involved. To show the sharpness, we consider the functions f and g given by f (z ) =

1 z

+

(B − A)(m! − α) (c )p+1 p ∗   z ∈ Σm (a + 1, c , A, B, α) (a + 1)p+1 p m! (1 + B)

(5.14)

m

and

 g (z ) =

 0

 (B − A)(m! − α) (c )p+1 (B − A)(m! − α)δ (c )p+1   zp,     + . +  z (a + 1)p+1 (a)p+1  p p m! (1 + B) m! (1 + B)

1

m

0

(5.15)

m

where δ > δ = a+p+1 . Clearly, the function g belongs to N +0 (f ). On the other hand, we find from Theorem 1 that g is not in δ ∗ the class Σm (a, c , A, B, α). Thus the proof of Theorem 5 is completed.  p+1

6. Convolution properties For the functions fj (z ) =

1 z

+

∞ X an,j z n (j = 1, 2; p ∈ N ∗ ) n =p

(6.1)

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M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

we denote by (f1 ∗ f2 ) the Hadamard product (or convolution) of the functions f1 and f2 , that is,

(f1 ∗ f2 )(z ) =

1 z

+

∞ X an,1 an,2 z n .

(6.2)

n =p

Throughout this section, we assume further that a > c > 0. ∗ Theorem 6. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σm (a, c , A, B, α). Then (f1 ∗ f2 ) ∈ Σm∗ (a, c , A, B, ζ ), where

ζ = m! −

(B − A)(m! − α)2 (c )p+1   . p m! (1 + B)(a)p+1

(6.3)

m

The result is sharp for the functions fj (j = 1, 2) given by fj (z ) =

1 z

+

(B − A)(m! − α)(c )p+1 p   z (j = 1, 2; p, m ∈ N ∗ ; p ≥ m). p m! (1 + B)(a)p+1

(6.4)

m

Proof. Employing the technique used earlier by Schild and Silverman [16], we need to find the largest ζ such that

  ∞ X n=p

n (1 + B)(a)n+1 m

m!

(B − A)(m! − ζ )(c )n+1

an,1 an,2 ≤ 1

(6.5)

∗ for fj ∈ Σm (a, c , A, B, α) (j = 1, 2). Since fj ∈ Σm∗ (a, c , A, B, α) (j = 1, 2), we readily see that

  n

∞ m! m (1 + B)(a)n+1 X an,j ≤ 1 (j = 1, 2). (B − A)(m! − α)(c )n+1 n=p

(6.6)

Therefore, by the Cauchy–Schwarz inequality, we obtain

 

n ∞ m! m (1 + B)(a)n+1 q X an,1 an,2 ≤ 1. (B − A)(m! − α)(c )n+1 n=p

(6.7)

This implies that we only need to show that 1

(m! − ζ )

an,1 an,2 ≤

1

(m! − α)

q an,1 an,2 (n ≥ p ≥ m)

(6.8)

or, equivalently, that

q an,1 an,2 ≤ (m! − ζ ) (n ≥ p ≥ m). (m! − α)

(6.9)

Hence, by the inequality (6.7), it is sufficient to prove that

(m! − ζ ) (B − A)(m! − α)(c )n+1   ≤ (n ≥ p ≥ m). (m! − α) n m! (1 + B)(a)n+1

(6.10)

m

It follows from (6.10) that

ζ ≤ m! −

(B − A)(m! − α)2 (c )n+1   n m! (1 + B)(a)n+1

(n ≥ p ≥ m).

m

Now, defining the function ϕ(n) by

ϕ(n) = m! −

(B − A)(m! − α)2 (c )n+1   n m! (1 + B)(a)n+1 m

(n ≥ p ≥ m).

(6.11)

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

877

But

(B − A)(m! − α)2 (c )n+1   ϕ(n + 1) − ϕ(n) = n m! (1 + B)(a)n+1



(n + 1)(a − c + m) + mc (n + 1)(a + n + 1)



>0

m

for a > c > 0, −1 ≤ A < B ≤ 1, 0 < B ≤ 1, 0 ≤ α < m! and p ≥ m. Then we see that ϕ(n) is an increasing function of n (n ≥ p ≥ m). Therefore, we conclude that

ζ ≤ ϕ(p) = m! −

(B − A)(m! − α)2 (c )p+1   , p m! (1 + B)(a)p+1

(6.12)

m

which evidently completes the proof of Theorem 6.



Putting (i) m = p ∈ N ∗ , a = c , A = −β and B = β (0 < β ≤ 1) (ii) m = p ∈ N ∗ , a = c , A = −β and B = (2γ − 1)β (0 < β ≤ 1 and 0 ≤ γ ≤ 12 ) in Theorem 6, we obtain the following consequences. Corollary 4. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σp∗ (α, β). Then (f1 ∗ f2 ) ∈ Σp∗ (ζ , β), where

ζ = p! −

2β(p! − α)2 p!(1 + β)

.

(6.13)

The result is sharp for the functions fj (j = 1, 2) given by fj (z ) =

1 z

+

2β(p! − α) p!(1 + β)

(j = 1, 2; p ∈ N ∗ ).

zp

(6.14)

Corollary 5. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σp∗ (α, β, γ ). Then (f1 ∗ f2 ) ∈ Σp∗ (ζ , β, γ ), where

ζ = p! −

2βγ (p! − α)2 p!(1 + 2βγ − β)

.

(6.15)

The result is sharp for the functions fj (j = 1, 2) given by fj (z ) =

1 z

+

2βγ (p! − α) p!(1 + 2βγ − β)

zp

(j = 1, 2; p ∈ N ∗ ).

(6.16)

Using arguments similar to those in the proof of Theorem 6, we obtain the following result: ∗ Theorem 7. Let the function f1 defined by (6.1) be in the class Σm (a, c , A, B, α1 ). Suppose also that the function f2 defined by ∗ ∗ (6.1) be in the class Σm (a, c , A, B, α2 ). Then (f1 ∗ f2 ) ∈ Σm (a, c , A, B, τ ), where

τ = m! −

(B − A)(m! − α1 )(m! − α2 )(c )p+1   . p m! (1 + B)(a)p+1

(6.17)

m

The result is sharp for the functions fj (j = 1, 2) given by f1 ( z ) =

1 z

+

(B − A)(m! − α1 )(c )p+1 p   z (p, m ∈ N ∗ ; p ≥ m) p m! (1 + B)(a)p+1

(6.18)

(B − A)(m! − α2 )(c )p+1 p   z (p, m ∈ N ∗ ; p ≥ m). p (1 + B)(a)p+1 m!

(6.19)

m

and f2 ( z ) =

1 z

+

m

Putting (i) m = p ∈ N ∗ , a = c , A = −β and B = β(0 < β ≤ 1) (ii) m = p ∈ N ∗ , a = c , A = −β and B = (2γ − 1)β(0 < β ≤ 1 and 12 ≤ γ ≤ 1) in Theorem 7, we obtain:

878

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

Corollary 6. Let the function f1 defined by (6.1) be in the class Σp∗ (α1 , β). Suppose also that the function f2 defined by (6.1) be in the class Σp∗ (α2 , β). Then (f1 ∗ f2 ) ∈ Σp∗ (τ , β), where 2β(p! − α1 )(p! − α2 )

τ = p! −

p!(1 + β)

.

(6.20)

The result is the best possible for the functions fj (j = 1, 2) given by f1 ( z ) =

1

f2 ( z ) =

1

2β(p! − α1 )

+

z

p!(1 + β)

zp

(p ∈ N ∗ )

(6.21)

zp

(p ∈ N ∗ ).

(6.22)

and 2β(p! − α2 )

+

z

p!(1 + β)

Corollary 7. Let the function f1 defined by (6.1) be in the class Σp∗ (α1 , β, γ ). Suppose also that the function f2 defined by (6.1) be in the class Σp∗ (α2 , β, γ ). Then (f1 ∗ f2 ) ∈ Σp∗ (τ , β, γ ), where 2βγ (p! − α1 )(p! − α2 )

τ = p! −

p!(1 + 2βγ − β)

.

(6.23)

The result is the best possible for the functions fj (j = 1, 2) given by f1 ( z ) =

1

f2 ( z ) =

1

2βγ (p! − α1 )

+

z

p!(1 + 2βγ − β)

zp

(p ∈ N ∗ )

(6.24)

zp

(p ∈ N ∗ ).

(6.25)

and 2βγ (p! − α2 )

+

z

p!(1 + 2βγ − β)

∗ Theorem 8. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σm (a, c , A, B, α). Then the function h defined by

h(z ) =

1

∞  X  an,1 2 + an,2 2 z n

+

z

(6.26)

n =p

∗ belongs to the class Σm (a, c , A, B, ξ ), where

2(B − A)(m! − α)2 (c )p+1

ξ = m! −

 

p m! (1 + B)(a)p+1 m

.

(6.27)

The result is sharp for the functions fj (j = 1, 2) defined by (6.4). Proof. Noting that

2

 



n ∞  m! m (1 + B)(a)n+1  X n=p

 

 ∞ X

  2   (B − A)(m! − α)(c )n+1  an,j ≤ 

n=p

m!

n (1 + B)(a)n+1 m

(B − A)(m! − α)(c )n+1

2

 an,j  ≤ 1 (j = 1, 2), 

(6.28)

∗ for fj ∈ Σm (a, c , A, B, α) (j = 1, 2), we have

 ∞ X 1 n=p

 

m!

n (1 + B)(a)n+1 m

2

2 [(B − A)(m! − α)(c )n+1 ]

2

  an,1 2 + an,2 2 ≤ 1.

(6.29)

Therefore, we have to find the largest ξ such that

  1

(m! − ξ )

m!

≤

n (1 + B)(a)n+1 m

2(B − A)(m! − α)2 (c )n+1

(n ≥ p ≥ m),

(6.30)

M.K. Aouf, R.M. El-Ashwah / Mathematical and Computer Modelling 49 (2009) 868–879

879

that is, that

ξ ≤ m! −

2(B − A)(m! − α)2 (c )n+1

(n ≥ p ≥ m).

  m!

n (1 + B)(a)n+1 m

(6.31)

Now, defining a function Ψ (n) by

Ψ (n) = m! −

2(B − A)(m! − α)2 (c )n+1

 

n m! (1 + B)(a)n+1 m

(n ≥ p ≥ m).

(6.32)

We observe that Ψ (n) is an increasing function of n (n ≥ p ≥ m). Thus, we conclude that

ξ ≤ Ψ (p) = m! −

2(B − A)(m! − α)2 (c )p+1

 

p (1 + B)(a)p+1 m

m!

which completes the proof of Theorem 8.

,

(6.33)



Putting (i) m = p ∈ N , a = c , A = −β and B = β (0 < β ≤ 1) (ii) m = p ∈ N ∗ , a = c , A = −β and B = (2γ − 1)β(0 < β ≤ 1 and 12 ≤ γ ≤ 1) in Theorem 8, we obtain: ∗

Corollary 8. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σp∗ (α, β), then the function h defined by (6.26) belongs to the class Σp∗ (ξ , β), where

ξ = p! −

4β (p! − α)2 p!(1 + β)

.

(6.34)

The result is sharp for the functions fj (j = 1, 2) defined by (6.14). Corollary 9. Let the functions fj (j = 1, 2) defined by (6.1) be in the class Σp∗ (α, β, γ ), then the function h defined by (6.26) belongs to the class Σp∗ (ξ , β, γ ), where

ξ = p! −

4βγ (p! − α)2 p!(1 + 2βγ − β)

.

(6.35)

The result is sharp for the functions fj (j = 1, 2) defined by (6.16). Acknowledgements The authors like to thank the referees for their valuable suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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