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On some applications of a subordination theorem
J. Math. Anal. Appl. 311 (2005) 489–494 www.elsevier.com/locate/jmaa
On some applications of a subordination theorem A.A. Attiya ∗ Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt Received 28 April 2004 Available online 30 March 2005 Submitted by J. Noguchi
Abstract The main object of the present paper is to show some interesting relations for certain class of analytic functions by using subordination theorem. 2005 Elsevier Inc. All rights reserved. Keywords: Analytic; Univalent; Prestarlike functions; Convex; Subordinating factor; Ruscheweyh derivative; Hadamard product
1. Introduction Let A denote the class of functions of the form ∞ an z n , f (z) = z +
(1.1)
n=2
which are analytic in the unit disc U = {z: |z| < 1}. Also let K denote (as usual) the class of functions f (z) ∈ A which are convex in U . Given two functions f, g ∈ A, where f (z) is given by (1.1) and g(z) is defined by g(z) = z +
∞
bn z n .
n=2
* Current address: Department of Mathematics, Teachers’ college in Abha, Abha 249, Saudi Arabia.
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.02.056
(1.2)
490
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
The Hadamard product f ∗ g(z) is defined by f ∗ g(z) = z +
∞
an bn z n ,
z ∈ U.
(1.3)
n=2
By using the Hadamard product, Ruscheweyh [4] defined z ∗ f (z) (α −1). D α f (z) = (1 − z)α+1
(1.4)
Ruscheweyh [4] observed that D k f (z) =
z(zk−1 f (z))(k) k!
(1.5)
when k = α ∈ N0 = N ∪ {0} = {0, 1, 2, . . .}. This symbol D k f (z) (k ∈ N0 ) was called the kth order Ruscheweyh derivative of f (z) by Al-Amiri [2]. We note that D 0 f (z) = f (z) and D 1 f (z) = zf (z). Let Rα denote the class of functions f (z) in A that satisfy the condition Re
D α+1 f (z) 1 > , D α f (z) 2
α −1, z ∈ U.
(1.6)
Ruscheweyh [5] called this class prestarlike of order α, see also Al-Amiri [1]. Also let Rα (β) denote the class of functions f (z) in A that satisfy the relation Re
D α+1 f (z) α + 2β > , D α f (z) 2(α + 1)
α 0, 0 β < 1, z ∈ U.
(1.7)
Al-Amiri [3] called this class prestarlike of order α and type β. Al-Amiri [3] proved that if ∞ (2n + α − 2β)C(α, n)|an | 2 + α − 2β, (1.8) n=2
then f (z) ∈ Rα (β), f (z) is defined by (1.1), α 0, 0 β < 1, and n (k + α − 1) , n = 2, 3, . . . . C(α, n) = k=2 (n − 1)!
(1.9)
Let Hα (β) denote the class of functions f (z) in A whose coefficients satisfy the condition (1.8). We note that Hα (β) ⊆ Rα (β). We easily see that Example 1.1. z ∈ Rα (α > −1); (i) f (z) = (1−z) (ii) f (z) = z 2 F 1 (1, 2 + α − 2β; 1 + α; z) ∈ Rα (β) (α 0, 0 β < 1); where ∞ (α)k (β)k k z 2 F 1 (α, β; γ ; z) = (γ )k k! k=0
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
491
is the hypergeometric function, and (µ)k is the Pochhammer symbol defined by 1, if k = 0, (µ)k = µ(µ + 1)(µ + 2) · · · (µ + k − 1), if k ∈ N. (iii) For α 0 and 0 β < 1, the functions 2 + α − 2β z2 , (1 + α)(4 + α − 2β) 2(2 + α − 2β) f2 (z) = z ± z3 and (1 + α)(2 + α)(6 + α − 2β) 1 3 + α − 4β f3 (z) = z ± z2 ± z3 2(4 + α − 2β) (1 + α)(2 + α)(6 + α − 2β) are members in the class Hα (β). f1 (z) = z ±
We need the following definitions and lemma. Definition 1.1 (Subordination Principle). Let g(z) be analytic and univalent in U . If f (z) is analytic in U , f (0) = g(0), and f (U ) ⊂ g(U ), then we say that f (z) is subordinate to g(z) in U , and we write f (z) ≺ g(z). We also say that g(z) is superordinate to f (z) in U . Definition 1.2 (Subordinating Factor Sequence). A sequence {bn }∞ n=1 of complex numbers is called a subordinating factor sequence if for every univalent function f (z) in K, we have the subordination given by ∞
bn an zn ≺ f (z)
(z ∈ U, a1 = 1).
(1.10)
n=1
Lemma 1.1. The sequence {bn }∞ n=1 is a subordinating factor sequence if and only if ∞ bn zn > 0 (z ∈ U ). Re 1 + 2 (1.11) n=1
This lemma due to Wilf [7].
2. Main theorem Theorem 2.1. Let f (z) ∈ Hα (β). Then (4 + α − 2β)(1 + α) (f ∗ g)(z) ≺ g(z) 2[α + (2 + α)(3 + α − 2β)] for every function g(z) in K, and [α + (2 + α)(3 + α − 2β)] Re f (z) > − . (4 + α − 2β)(1 + α) The constant
(4+α−2β)(1+α) 2[α+(2+α)(3+α−2β)]
is the best estimate.
(z ∈ U )
(2.1)
(2.2)
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A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
n Proof. Let f (z) ∈ Hα (β) and let g(z) = z + ∞ n=2 cn z be any function in the class K. Then (4 + α − 2β)(1 + α) (f ∗ g)(z) 2[α + (2 + α)(3 + α − 2β)] ∞ (4 + α − 2β)(1 + α) = an cn z n . (2.3) z+ 2[α + (2 + α)(3 + α − 2β)] n=2
Thus, by Definition 1.2, (2.1) will hold if the sequence the assertion of the theorem will hold if the sequence ∞ (4 + α − 2β)(1 + α)an (2.4) 2[α + (2 + α)(3 + α − 2β)] n=1 is a subordinating factor sequence, with a1 = 1. In view of Lemma 1.1, this equivalent to ∞ (4 + α − 2β)(1 + α) n an z > 0 (z ∈ U ). Re 1 + 2 (2.5) 2[α + (2 + α)(3 + α − 2β)] n=1
Now
∞
(4 + α − 2β)(1 + α) Re 1 + an z n [α + (2 + α)(3 + α − 2β)] n=1 (4 + α − 2β)(1 + α) z = Re 1 + [α + (2 + α)(3 + α − 2β)]
∞
1 + (4 + α − 2β)(1 + α)an zn [α + (2 + α)(3 + α − 2β)] n=2 (4 + α − 2β)(1 + α) >1− r [α + (2 + α)(3 + α − 2β)]
∞ 1 n (2n + α − 2β)C(α, n)|an |r − [α + (2 + α)(3 + α − 2β)] n=2
because C(α, n) is increasing function of n (4 + α − 2β)(1 + α) 2 + α − 2β >1− r− r (|z| = r) [α + (2 + α)(3 + α − 2β)] [α + (2 + α)(3 + α − 2β)] > 0. Thus (2.5) holds true in U . This proves (2.1), (2.2) follows by taking g(z) = Now, we consider the function 2 + α − 2β z2 (α 0, 0 β < 1), f0 (z) = z − (4 + α − 2β)(1 + α) which is a member of the class Hα (β); then by using (2.1), we have z (4 + α − 2β)(1 + α) f0 (z) ≺ . 2[α + (2 + α)(3 + α − 2β)] 1−z
z 1−z
in (2.1). (2.6)
(2.7)
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
493
It can be easily verified that min Re
1 (4 + α − 2β)(1 + α) f0 (z) =− 2[α + (2 + α)(3 + α − 2β)] 2
(z ∈ U ),
(2.8)
(4+α−2β)(1+α) then, the constant 2[α+(2+α)(3+α−2β)] cannot be replaced by any larger one. The proof is complete. 2
Taking β = 12 , we obtain the following corollary: Corollary 2.1. Let f (z) ∈ Rα , then (3 + α) (f ∗ g)(z) ≺ g(z) 2(4 + α)
(z ∈ U, α 0)
(2.9)
for every function g(z) in K, and Re f (z) > − The constant
(4 + α) . (3 + α)
(3+α) 2(4+α)
(2.10)
is the best estimate.
Taking α = 0 and β = 0 in the Theorem 2.1, we obtain the following corollary: Corollary 2.2 [6]. If the function f (z) of the form (1.1) is regular in U and satisfies ∞ n|a | n 1, then for every function g(z) in K, we have n=2 1 (f ∗ g)(z) ≺ g(z) 3
|z| < 1 ,
in particular Re{f (z)} > − 23 , z ∈ U . The constant
1 3
cannot be replaced by any larger one.
Acknowledgment The author thanks the referee for his helpful suggestions.
References [1] H.S. Al-Amiri, Certain generalization of prestarlike functions, J. Austral. Math. Soc. Ser. A 19 (1979) 325– 334, MR0557283 (81b:30018). [2] H.S. Al-Amiri, On Ruscheweyh derivatives, Ann. Polon. Math. 38 (1980) 87–94, MR0595351 (82c:30010). [3] H.S. Al-Amiri, Prestarlike functions of order α and type β with negative coefficients, Ann. Univ. Mariae Curie-Skłodowska Sect. A 39 (1985) 1–11, MR0997748 (90i:30017). [4] St. Ruscheweyh, New Criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975) 109–115, MR0367176 (51 #3418). [5] St. Ruscheweyh, Linear operator between classes of starlike functions, Comment. Math. Helv. 52 (1977) 497–509, MR0480976 (58 #1123).
494
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
[6] S. Singh, A subordination theorem for spirallike functions, Int. J. Math. Math. Sci. 24 (7) (2000) 433–435, MR1781509 (2001g:30014). [7] H.S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc. 12 (1961) 689–693, MR0125214 (23 #A2519).
On some applications of a subordination theorem A.A. Attiya ∗ Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 35516, Egypt Received 28 April 2004 Available online 30 March 2005 Submitted by J. Noguchi
Abstract The main object of the present paper is to show some interesting relations for certain class of analytic functions by using subordination theorem. 2005 Elsevier Inc. All rights reserved. Keywords: Analytic; Univalent; Prestarlike functions; Convex; Subordinating factor; Ruscheweyh derivative; Hadamard product
1. Introduction Let A denote the class of functions of the form ∞ an z n , f (z) = z +
(1.1)
n=2
which are analytic in the unit disc U = {z: |z| < 1}. Also let K denote (as usual) the class of functions f (z) ∈ A which are convex in U . Given two functions f, g ∈ A, where f (z) is given by (1.1) and g(z) is defined by g(z) = z +
∞
bn z n .
n=2
* Current address: Department of Mathematics, Teachers’ college in Abha, Abha 249, Saudi Arabia.
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.02.056
(1.2)
490
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
The Hadamard product f ∗ g(z) is defined by f ∗ g(z) = z +
∞
an bn z n ,
z ∈ U.
(1.3)
n=2
By using the Hadamard product, Ruscheweyh [4] defined z ∗ f (z) (α −1). D α f (z) = (1 − z)α+1
(1.4)
Ruscheweyh [4] observed that D k f (z) =
z(zk−1 f (z))(k) k!
(1.5)
when k = α ∈ N0 = N ∪ {0} = {0, 1, 2, . . .}. This symbol D k f (z) (k ∈ N0 ) was called the kth order Ruscheweyh derivative of f (z) by Al-Amiri [2]. We note that D 0 f (z) = f (z) and D 1 f (z) = zf (z). Let Rα denote the class of functions f (z) in A that satisfy the condition Re
D α+1 f (z) 1 > , D α f (z) 2
α −1, z ∈ U.
(1.6)
Ruscheweyh [5] called this class prestarlike of order α, see also Al-Amiri [1]. Also let Rα (β) denote the class of functions f (z) in A that satisfy the relation Re
D α+1 f (z) α + 2β > , D α f (z) 2(α + 1)
α 0, 0 β < 1, z ∈ U.
(1.7)
Al-Amiri [3] called this class prestarlike of order α and type β. Al-Amiri [3] proved that if ∞ (2n + α − 2β)C(α, n)|an | 2 + α − 2β, (1.8) n=2
then f (z) ∈ Rα (β), f (z) is defined by (1.1), α 0, 0 β < 1, and n (k + α − 1) , n = 2, 3, . . . . C(α, n) = k=2 (n − 1)!
(1.9)
Let Hα (β) denote the class of functions f (z) in A whose coefficients satisfy the condition (1.8). We note that Hα (β) ⊆ Rα (β). We easily see that Example 1.1. z ∈ Rα (α > −1); (i) f (z) = (1−z) (ii) f (z) = z 2 F 1 (1, 2 + α − 2β; 1 + α; z) ∈ Rα (β) (α 0, 0 β < 1); where ∞ (α)k (β)k k z 2 F 1 (α, β; γ ; z) = (γ )k k! k=0
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
491
is the hypergeometric function, and (µ)k is the Pochhammer symbol defined by 1, if k = 0, (µ)k = µ(µ + 1)(µ + 2) · · · (µ + k − 1), if k ∈ N. (iii) For α 0 and 0 β < 1, the functions 2 + α − 2β z2 , (1 + α)(4 + α − 2β) 2(2 + α − 2β) f2 (z) = z ± z3 and (1 + α)(2 + α)(6 + α − 2β) 1 3 + α − 4β f3 (z) = z ± z2 ± z3 2(4 + α − 2β) (1 + α)(2 + α)(6 + α − 2β) are members in the class Hα (β). f1 (z) = z ±
We need the following definitions and lemma. Definition 1.1 (Subordination Principle). Let g(z) be analytic and univalent in U . If f (z) is analytic in U , f (0) = g(0), and f (U ) ⊂ g(U ), then we say that f (z) is subordinate to g(z) in U , and we write f (z) ≺ g(z). We also say that g(z) is superordinate to f (z) in U . Definition 1.2 (Subordinating Factor Sequence). A sequence {bn }∞ n=1 of complex numbers is called a subordinating factor sequence if for every univalent function f (z) in K, we have the subordination given by ∞
bn an zn ≺ f (z)
(z ∈ U, a1 = 1).
(1.10)
n=1
Lemma 1.1. The sequence {bn }∞ n=1 is a subordinating factor sequence if and only if ∞ bn zn > 0 (z ∈ U ). Re 1 + 2 (1.11) n=1
This lemma due to Wilf [7].
2. Main theorem Theorem 2.1. Let f (z) ∈ Hα (β). Then (4 + α − 2β)(1 + α) (f ∗ g)(z) ≺ g(z) 2[α + (2 + α)(3 + α − 2β)] for every function g(z) in K, and [α + (2 + α)(3 + α − 2β)] Re f (z) > − . (4 + α − 2β)(1 + α) The constant
(4+α−2β)(1+α) 2[α+(2+α)(3+α−2β)]
is the best estimate.
(z ∈ U )
(2.1)
(2.2)
492
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
n Proof. Let f (z) ∈ Hα (β) and let g(z) = z + ∞ n=2 cn z be any function in the class K. Then (4 + α − 2β)(1 + α) (f ∗ g)(z) 2[α + (2 + α)(3 + α − 2β)] ∞ (4 + α − 2β)(1 + α) = an cn z n . (2.3) z+ 2[α + (2 + α)(3 + α − 2β)] n=2
Thus, by Definition 1.2, (2.1) will hold if the sequence the assertion of the theorem will hold if the sequence ∞ (4 + α − 2β)(1 + α)an (2.4) 2[α + (2 + α)(3 + α − 2β)] n=1 is a subordinating factor sequence, with a1 = 1. In view of Lemma 1.1, this equivalent to ∞ (4 + α − 2β)(1 + α) n an z > 0 (z ∈ U ). Re 1 + 2 (2.5) 2[α + (2 + α)(3 + α − 2β)] n=1
Now
∞
(4 + α − 2β)(1 + α) Re 1 + an z n [α + (2 + α)(3 + α − 2β)] n=1 (4 + α − 2β)(1 + α) z = Re 1 + [α + (2 + α)(3 + α − 2β)]
∞
1 + (4 + α − 2β)(1 + α)an zn [α + (2 + α)(3 + α − 2β)] n=2 (4 + α − 2β)(1 + α) >1− r [α + (2 + α)(3 + α − 2β)]
∞ 1 n (2n + α − 2β)C(α, n)|an |r − [α + (2 + α)(3 + α − 2β)] n=2
because C(α, n) is increasing function of n (4 + α − 2β)(1 + α) 2 + α − 2β >1− r− r (|z| = r) [α + (2 + α)(3 + α − 2β)] [α + (2 + α)(3 + α − 2β)] > 0. Thus (2.5) holds true in U . This proves (2.1), (2.2) follows by taking g(z) = Now, we consider the function 2 + α − 2β z2 (α 0, 0 β < 1), f0 (z) = z − (4 + α − 2β)(1 + α) which is a member of the class Hα (β); then by using (2.1), we have z (4 + α − 2β)(1 + α) f0 (z) ≺ . 2[α + (2 + α)(3 + α − 2β)] 1−z
z 1−z
in (2.1). (2.6)
(2.7)
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
493
It can be easily verified that min Re
1 (4 + α − 2β)(1 + α) f0 (z) =− 2[α + (2 + α)(3 + α − 2β)] 2
(z ∈ U ),
(2.8)
(4+α−2β)(1+α) then, the constant 2[α+(2+α)(3+α−2β)] cannot be replaced by any larger one. The proof is complete. 2
Taking β = 12 , we obtain the following corollary: Corollary 2.1. Let f (z) ∈ Rα , then (3 + α) (f ∗ g)(z) ≺ g(z) 2(4 + α)
(z ∈ U, α 0)
(2.9)
for every function g(z) in K, and Re f (z) > − The constant
(4 + α) . (3 + α)
(3+α) 2(4+α)
(2.10)
is the best estimate.
Taking α = 0 and β = 0 in the Theorem 2.1, we obtain the following corollary: Corollary 2.2 [6]. If the function f (z) of the form (1.1) is regular in U and satisfies ∞ n|a | n 1, then for every function g(z) in K, we have n=2 1 (f ∗ g)(z) ≺ g(z) 3
|z| < 1 ,
in particular Re{f (z)} > − 23 , z ∈ U . The constant
1 3
cannot be replaced by any larger one.
Acknowledgment The author thanks the referee for his helpful suggestions.
References [1] H.S. Al-Amiri, Certain generalization of prestarlike functions, J. Austral. Math. Soc. Ser. A 19 (1979) 325– 334, MR0557283 (81b:30018). [2] H.S. Al-Amiri, On Ruscheweyh derivatives, Ann. Polon. Math. 38 (1980) 87–94, MR0595351 (82c:30010). [3] H.S. Al-Amiri, Prestarlike functions of order α and type β with negative coefficients, Ann. Univ. Mariae Curie-Skłodowska Sect. A 39 (1985) 1–11, MR0997748 (90i:30017). [4] St. Ruscheweyh, New Criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975) 109–115, MR0367176 (51 #3418). [5] St. Ruscheweyh, Linear operator between classes of starlike functions, Comment. Math. Helv. 52 (1977) 497–509, MR0480976 (58 #1123).
494
A.A. Attiya / J. Math. Anal. Appl. 311 (2005) 489–494
[6] S. Singh, A subordination theorem for spirallike functions, Int. J. Math. Math. Sci. 24 (7) (2000) 433–435, MR1781509 (2001g:30014). [7] H.S. Wilf, Subordinating factor sequences for convex maps of the unit circle, Proc. Amer. Math. Soc. 12 (1961) 689–693, MR0125214 (23 #A2519).