Báo cáo toán học: " Subclasses of Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator"

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Nội dung Text: Báo cáo toán học: " Subclasses of Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator"

9LHWQDP -RXUQDO Vietnam Journal of Mathematics 33:3 (2005) 319–334 RI 0$7+(0$7,&6 9$67 Subclasses of Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator Maslina Darus1 , Aini Janteng2 , and Suzeini Abdul Halim2 1 School of Mathematical Sciences, Faculty of Sciences and Technology, University Kebangsaan Malaysia 43600 Bangi, Selangor, Malaysia 2 Institute of Mathematical Sciences, University Malaya 50603 Kuala Lumpur, Malaysia Received July 21, 2004 Revised March 2, 2005 Abstract. In this paper, we consider a class of uniformly starlike functions deﬁned by certain integral operator. We determine a suﬃcient condition for a function f to be uniformly starlike function that is also necessary when f has negative coeﬃcients. Similar results for corresponding subclasses of uniformly convex functions are also obtained. 1. Introduction Let S denote the class of functions f which are analytic and univalent in D = {z : 0 < |z | < 1} and given by ∞ an z n , an ≥ 0 . f (z ) = z + (1) n=2 A function f ∈ S is called a uniformly starlike function if and only if z f (z ) z f (z ) ≥ −1 , z ∈ D. Re f (z ) f (z ) We denote this class by Sp . A function f ∈ S is called a uniformly convex function if and only if
320 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim zf (z ) z f (z ) ≥ , z ∈ D. Re 1 + f (z ) f (z ) We denote this class by UCV . Rønning [3] generalized the class Sp and UCV by introducing a parameter α in the following way. Deﬁnition 1. [4] A function f ∈ Sp (α), 0 α 1, if f satisﬁes the analytic characterization z f (z ) z f (z ) −1 −α Re f (z ) f (z ) and f ∈ UCV (α) if and only if zf ∈ Sp (α). In [1], Bharati et al. obtained coeﬃcient characterization for some subclasses of Sp (α) and UCV (α). Deﬁnition 2. [5] Let T be the subclass of S consisting of functions f of the form ∞ an z n , an ≥ 0 . f (z ) = z − (2) n=2 Also, Bharati et al. in [1] obtained coeﬃcient characterization for some subclasses of Sp (α) and UCV (α) for f ∈ T . Recently, Jung et al. [2] introduced the following one-parameter families of integral operator for functions f ∈ S : z α−1 α+β α t Qα f (z ) tβ −1 f (t)dt, (α > 0, β > −1) 1− = (3) β zβ β z 0 and z α+1 tα−1 f (t)dt, (α > −1). Jα f (z ) = (4) zα 0 They showed that ∞ Γ(α + β + 1) Γ(β + n) Qα f (z ) = z + an z n , (α > 0, β > −1) (5) β Γ(β + 1) n=2 Γ(α + β + n) and ∞ α+1 an z n , (α > −1). Jα f (z ) = z + (6) α+n n=2
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 321 By virtue (5) and (6), we see that Jα f (z ) = Q1 f (z ), (α > −1). (7) α For f ∈ T , the operator in (5) and (6) becomes ∞ Γ(α + β + 1) Γ(β + n) Qα f (z ) = z − an z n , (α > 0, β > −1) (8) β Γ(β + 1) n=2 Γ(α + β + n) and ∞ α+1 an z n , (α > −1). Jα f (z ) = z − (9) α+n n=2 Using equations (3) and (4), we introduce the following new subclasses of Sp (α). Deﬁnition 3. Let T Q(α, β, σ ), α > 0, β > −1 and 0 σ 1 be the class of functions f ∈ T satisfying the condition z (Qα f (z )) z (Qα f (z )) β β −1 − σ, z∈D Re (10) Qα f (z ) Qα f (z ) β β where Qα f is deﬁned as in (3). β Deﬁnition 4. Let T J (α, σ ), α > −1 and 0 σ 1 be the class of functions f ∈ T satisfying the condition z (Jα f (z )) z (Jα f (z )) −1 − σ, z∈D Re (11) Jα f (z ) Jα f (z ) where Jα f is deﬁned as in (4). 2. Properties of T Q(α, β, σ ) In this section, we give some results for T Q(α, β, σ ). We ﬁrst state a preliminary lemma, required for proving our result. Lemma 1. If Qα f ∈ T then β ∞ Γ(α + β + 1) Γ(β + n) nan ≤ 1. Γ(β + 1) Γ(α + β + n) n=2 ∞ Γ(α+β +1) Γ(β +n) Proof. Suppose on the contrary that nan > 1. We can n=2 Γ(α+β +n) Γ(β +1) write ∞ Γ(α + β + 1) Γ(β + n) nan = 1 + ε, (ε > 0). Γ(β + 1) Γ(α + β + n) n=2 Then there exists an integer N such that
322 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim N Γ(α + β + 1) Γ(β + n) ε nan > 1 + . Γ(β + 1) Γ(α + β + n) 2 n=2 1 1 N −1 For < z < 1, we have ε ∞ 1+ 2 Γ(α + β + 1) Γ(β + n) (Qα f (z )) = 1 − nan z n−1 β Γ(β + 1) Γ(α + β + n) n=2 N Γ(α + β + 1) Γ(β + n) nan z n−1 1− Γ(β + 1) Γ(α + β + n) n=2 N Γ(α + β + 1) N −1 Γ(β + n) 1− z nan Γ(β + 1) Γ(α + β + n) n=2 ε < 1 − z N −1 1 + 2 < 0. 1 1 N −1 Since (Qα f (0)) = 1 > 0, there exists a real number z0 , 0 < z0 < , β ε 1+ 2 such that (Qα f (z0 )) = 0. Hence Qα f is not univalent. β β Theorem 1. Let the functions f ∈ T . Then ∞ Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )an 1−σ (12) Γ(β + 1) n=2 Γ(α + β + n) for some α > 0, β > −1 and 0 1 if and only if f ∈ T Q(α, β, σ ). σ Proof. First, we have z (Qα f (z )) z (Qα f (z )) z (Qα f (z )) β β β − 1 − Re −1 −1 2 Qα f (z ) Qα f (z ) Qα f (z ) β β β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) 2(n − 1)|an ||z | Γ(β +1) ∞ 1 − Γ(α++1)β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) |an ||z | Γ(β ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) 2(n − 1)an Γ(β +1) , ∞ 1 − Γ(α++1)β +1) Γ(β +n) n=2 Γ(α+β +n) an Γ(β where ∞ Γ(α + β + 1) Γ(β + n) 1− |an | > 0 Γ(β + 1) n=2 Γ(α + β + n) by Lemma 1. The above expression is bounded by 1 − σ if and only if (12) is satisﬁed. Consequently, we can write z (Qα f (z )) z (Qα f (z )) β β − 1 − Re −1 1−σ Qα f (z ) Qα f (z ) β β
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 323 which is equivalent to (10). Conversely, if f ∈ T Q(α, β, σ ) and z is real, then Deﬁnition 3 yields ∞ Γ(α+β +1) Γ(β +n) n−1 1− n=2 Γ(α+β +n) nan z Γ(β +1) −σ ∞ Γ(α+β +1) Γ(β +n) n−1 − Γ(β +1) 1 n=2 Γ(α+β +n) an z ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) (n − 1)an z Γ(β +1) ≥ . ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) an z Γ(β Let z → 1− along the real axis, then we get ∞ ∞ Γ(α+β +1) Γ(β +n) Γ(α+β +1) Γ(β +n) 1− n=2 Γ(α+β +n) (n − 1)an n=2 Γ(α+β +n) nan Γ(β +1) Γ(β +1) − ≥σ ∞ ∞ Γ(α+β +1) Γ(β +n) 1 − Γ(α++1) β +1) Γ(β +n) − Γ(β +1) 1 n=2 Γ(α+β +n) an n=2 Γ(α+β +n) an Γ(β or ∞ Γ(α + β + 1) Γ(β + n) 1− (2n − 1)an Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) ≥σ 1− an Γ(β + 1) n=2 Γ(α + β + n) which gives the required result. Our assertion in Theorem 1 is sharp, for functions of the form Γ(β + 1) Γ(α + β + n) 1 − σ Fn (z ) = Qα fn (z ) = z − z n , n ≥ 2 (13) β Γ(α + β + 1) Γ(β + n) 2n − 1 − σ which belong to the class T Q(α, β, σ ). Corollary 1. If f ∈ T Q(α, β, σ ) then Γ(β + 1) Γ(α + β + n) 1 − σ , n ≥ 2. an (14) Γ(α + β + 1) Γ(β + n) 2n − 1 − σ Proof. Since f ∈ T Q(α, β, σ ), Theorem 1 gives ∞ Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )an 1 − σ. Γ(β + 1) n=2 Γ(α + β + n) Next, note that Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )an Γ(β + 1) Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )an . Γ(β + 1) n=2 Γ(α + β + n) Therefore Γ(β + 1) Γ(α + β + n) 1 − σ , n ≥ 2. an Γ(α + β + 1) Γ(β + n) 2n − 1 − σ
324 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim Corollary 2. If f ∈ T Q(α, β, σ ) and |z | = r < 1, then 1−σ 2 (i) |Qα f (z )| r + r β 3−σ 1−σ 2 (ii) |Qα f (z )| ≥ r − r. β 3−σ The results are sharp. Proof. First, it is obvious that ∞ Γ(α + β + 1) Γ(β + 2) (3 − σ ) an Γ(β + 1) Γ(α + β + 2) n=2 ∞ Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )an Γ(β + 1) n=2 Γ(α + β + n) and as f ∈ T Q(α, β, σ ), using the inequality in Theorem 1 yields ∞ Γ(β + 1) Γ(α + β + 2) 1 − σ an . (15) Γ(α + β + 1) Γ(β + 2) 3 − σ n=2 From (8) with |z | = r(r < 1), we have ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| an r n r+ β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 r+ Γ(β + 1) Γ(α + β + 2) n=2 and ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| ≥ r − an r n β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 . ≥r− Γ(β + 1) Γ(α + β + 2) n=2 Finally, using (15) in the above inequalities gives us both results (i) and (ii). We note that (i) and (ii) are sharp for the following function Γ(β + 1) Γ(α + β + 2) 1 − σ 2 Qα f2 (z ) = z − z β Γ(α + β + 1) Γ(β + 2) 3 − σ at z = ±ir, ±r. Deﬁnition 5. Let T Q(α, β, σ, γ ) be the class of functions f ∈ T satisfying the condition z (Qα f (z )) z (Qα f (z )) β β ≥σ −1 +γ, α > 0, β > −1, 0 Re σ 1, 0 γ 1 Qα f (z ) Qα f (z ) β β (16) where Qα f is deﬁned as in (3). β We write T Q(α, β, 1, γ ) = T Q(α, β, γ ).
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 325 Theorem 2. Let the functions f ∈ T . A function f ∈ T Q(α, β, σ, γ ) for some α > 0, β > −1, 0 σ 1 and 0 γ 1 if and only if ∞ Γ(α + β + 1) Γ(β + n) [n(1 + σ ) − (σ + γ )]an 1 − γ. (17) Γ(β + 1) n=2 Γ(α + β + n) Proof. In view of Deﬁnition 5, it suﬃces to show that z (Qα f (z )) z (Qα f (z )) β β −1 − γ. σ Re (18) Qα f (z ) Qα f (z ) β β The condition (18) is equivalent to z (Qα f (z )) z (Qα f (z )) β β − 1 − Re −1 1 − γ. σ Qα f (z ) Qα f (z ) β β Then z (Qα f (z )) z (Qα f (z )) z (Qα f (z )) β β β − 1 − Re −1 −1 σ (σ + 1) Qα f (z ) Qα f (z ) Qα f (z ) β β β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) (σ + 1)(n − 1)|an ||z | Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) n−1 1− n=2 Γ(α+β +n) |an ||z | Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) (σ + 1)(n − 1)an Γ(β +1) . ∞ Γ(α+β +1) Γ(β +n) 1− n=2 Γ(α+β +n) an Γ(β +1) The above expression is bounded by 1 − γ if and only if (17) is satisﬁed. Conversely, if f ∈ T Q(α, β, σ, γ ) and z is real, we get ∞ Γ(α+β +1) Γ(β +n) n−1 1− n=2 Γ(α+β +n) nan z Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) n−1 − Γ(β +1) 1 n=2 Γ(α+β +n) an z ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) (n − 1)an z Γ(β +1) ≥σ + γ. ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) an z Γ(β Let z → 1− along the real axis, which gives ∞ Γ(α+β +1) Γ(β +n) 1− n=2 Γ(α+β +n) nan Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) − Γ(β +1) 1 n=2 Γ(α+β +n) an ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) (n − 1)an Γ(β +1) ≥σ + γ, ∞ 1 − Γ(α++1) β +1) Γ(β +n) n=2 Γ(α+β +n) an Γ(β that is equivalent to ∞ Γ(α + β + 1) Γ(β + n) 1− (n + σn − σ )an Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) ≥γ 1− an Γ(β + 1) n=2 Γ(α + β + n)
326 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim and gives the required result. Then, the proof is complete. Our assertion in Theorem 2 is sharp for functions of the form 1−γ Γ(β + 1) Γ(α + β + n) Fn (z ) = Qα fn (z ) = z − zn, n ≥ 2 β Γ(α + β + 1) Γ(β + n) n(1 + σ ) − (σ + γ ) (19) which belong to the class T Q(α, β, σ, γ ). Corollary 3. If f ∈ T Q(α, β, σ, γ ) then 1−γ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an (20) Γ(α + β + 1) Γ(β + n) n(1 + σ ) − (σ + γ ) Proof. Since f ∈ T Q(α, β, σ, γ ), Theorem 2 yields ∞ Γ(α + β + 1) Γ(β + n) [n(1 + σ ) − (σ + γ )]an 1 − γ. Γ(β + 1) n=2 Γ(α + β + n) Next, note that Γ(α + β + 1) Γ(β + n) [n(1 + σ ) − (σ + γ )]an Γ(β + 1) Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) [n(1 + σ ) − (σ + γ )]an . Γ(β + 1) n=2 Γ(α + β + n) Therefore 1−γ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an Γ(α + β + 1) Γ(β + n) n(1 + σ ) − (σ + γ ) Corollary 4. If f ∈ T Q(α, β, σ, γ ) and |z | = r < 1, then 1−γ r2 , (i) |Qα f (z )| r + β 2+σ−γ 1−γ r2 . (ii) |Qα f (z )| ≥ r − β 2+σ−γ Proof. First, it is obvious that ∞ Γ(α + β + 1) Γ(β + 2) (2 + σ − γ ) an Γ(β + 1) Γ(α + β + 2) n=2 ∞ Γ(α + β + 1) Γ(β + n) [n(1 + σ ) − (σ + γ )]an , Γ(β + 1) n=2 Γ(α + β + n) and as f ∈ T Q(α, β, σ, γ ), using the inequality in Theorem 2 yields ∞ Γ(β + 1) Γ(α + β + 2) 1 − γ an . (21) Γ(α + β + 1) Γ(β + 2) 2 + σ − γ n=2 From (8) with |z | = r(r < 1), we have
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 327 ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| an r n r+ β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 r+ Γ(β + 1) Γ(α + β + 2) n=2 and ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| ≥ r − an r n β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 . ≥r− Γ(β + 1) Γ(α + β + 2) n=2 Finally, using (21) in the above inequalities gives us both results (i) and (ii). We note that (i) and (ii) are sharp for the following function Γ(β + 1) Γ(α + β + 2) 1 − γ Qα f2 (z ) = z − z2 β Γ(α + β + 1) Γ(β + 2) 2 + σ − γ at z = ±ir, ±r. Remark 1. By taking β = α and α = 1 in (8), analogous results for T J (α, σ ) are also obtained. 3. Properties of UCV QT (α, β, σ ) Now, let us draw our attention to the following new subclasses of UCV (α) and ﬁnd their coeﬃcient criterion. Deﬁnition 6. Let UCV QT (α, β, σ ), α > 0, β > −1 and 0 σ 1 be the class of functions f ∈ T which satisfy the condition z (Qα f (z )) z (Qα f (z )) β β −σ , z∈D Re 1+ (22) (Qα f (z )) (Qα f (z )) β β where Qα f is deﬁned as in (3). β Deﬁnition 7. Let UCV JT (α, σ ), α > −1 and 0 σ 1 be the class of functions f ∈ T which satisfy the condition z (Jα f (z )) z (Jα f (z )) − σ , z ∈ D, Re 1+ (23) (Jα f (z )) (Jα f (z )) where Jα f is deﬁned as in (4). Next, we give some results for UCV QT (α, β, σ ) as the following. Theorem 3. Let the functions f ∈ T . Then
328 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim ∞ Γ(α + β + 1) Γ(β + n) (2n − 1 − σ )nan 1−σ (24) Γ(β + 1) n=2 Γ(α + β + n) for some α > 0, β > −1 and 0 1 if and only if f ∈ UCV QT (α, β, σ ). σ Proof. First, we consider z (Qα f (z )) z (Qα f (z )) z (Qα f (z )) β β β − Re 2 (Qα f (z )) (Qα f (z )) (Qα f (z )) β β β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) 2n(n − 1)|an ||z | Γ(β +1) ∞ 1 − Γ(α++1)β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n|an ||z | Γ(β ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) 2n(n − 1)an Γ(β +1) , ∞ 1 − Γ(α++1)β +1) Γ(β +n) n=2 Γ(α+β +n) nan Γ(β ∞ where 1 − Γ(α++1) β +1) Γ(β +n) n=2 Γ(α+β +n) n|an | > 0 by getting use of Lemma 1. Γ(β The above expression is bounded by 1 − σ if and only if (24) is satisﬁed. Consequently, we can write z (Qα f (z )) z (Qα f (z )) β β + 1 − σ. Re (Qα f (z )) (Qα f (z )) β β which is equivalent to (22). Conversely, if f ∈ UCV QT (α, β, σ ) and z is real, then Deﬁnition 6 yields ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) 1− −σ ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) nan z Γ(β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) ≥ . ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) nan z Γ(β Let z → 1− along the real axis, then we get ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) 2n(n − 1)an Γ(β +1) (1 − σ ) ≥ ∞ 1 − Γ(α++1)β +1) Γ(β +n) n=2 Γ(α+β +n) nan Γ(β or ∞ Γ(α + β + 1) Γ(β + n) (1 − σ ) 1 − nan Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) ≥ 2n(n − 1)an , Γ(β + 1) n=2 Γ(α + β + n) which gives the required result. Our assertion in Theorem 3 is sharp for functions of the form
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 329 1−σ Γ(β + 1) Γ(α + β + n) Fn (z ) = Qα fn (z ) = z − z n, n≥2 β Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ ) (25) which belong to the class UCV QT (α, β, σ ). Corollary 5. If f ∈ UCV QT (α, β, σ ) then 1−σ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an (26) Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ ) Proof. Since f ∈ UCV QT (α, β, σ ), Theorem 3 gives ∞ Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ )an 1 − σ. Γ(β + 1) n=2 Γ(α + β + n) Next, note that Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ )an Γ(β + 1) Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ )an . Γ(β + 1) n=2 Γ(α + β + n) Therefore 1−σ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ ) Corollary 6. If f ∈ UCV QT (α, β, σ ) and |z | = r < 1, then 1−σ r2 , (i) |Qα f (z )| r + β 2(3 − σ ) 1−σ r2 . (ii) |Qα f (z )| ≥ r − β 2(3 − σ ) Proof. First, it is obvious that ∞ Γ(α + β + 1) Γ(β + 2) 2(3 − σ ) an Γ(β + 1) Γ(α + β + 2) n=2 ∞ Γ(α + β + 1) Γ(β + n) n(2n − 1 − σ )an , Γ(β + 1) n=2 Γ(α + β + n) and as f ∈ UCV QT (α, β, σ ), using the inequality in Theorem 3 yields ∞ Γ(β + 1) Γ(α + β + 2) 1 − σ an . (27) Γ(α + β + 1) Γ(β + 2) 2(3 − σ ) n=2 From (8) with |z | = r(r < 1), we have ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| an r n r+ β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 r+ Γ(β + 1) Γ(α + β + 2) n=2
330 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim and ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| an r n ≥r− β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 . ≥r− Γ(β + 1) Γ(α + β + 2) n=2 Finally, using (27) in the above inequalities gives us both results (i) and (ii). We note that (i) and (ii) are sharp for the following function Γ(β + 1) Γ(α + β + 2) 1 − σ Qα f2 (z ) = z − z2 β Γ(α + β + 1) Γ(β + 2) 2(3 − σ ) at z = ±ir, ±r. Deﬁnition 8. Let UCV QT (α, β, σ, γ ) be the class of functions f ∈ T satisfying the condition z (Qα f (z )) z (Qα f (z )) β β ≥σ +γ, α > 0, β > −1, 0 Re 1+ σ 1, 0 γ 1. (Qα f (z )) (Qα f (z )) β β (28) where Qα f is deﬁned as in (3). β We write UCV QT (α, β, 1, γ ) = UCV QT (α, β, γ ). Theorem 4. Let the functions f ∈ T . A function f ∈ UCV QT (α, β, σ, γ ) for some α > 0, β > −1, 0 σ 1 and 0 γ 1 if and only if ∞ Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )]an 1 − γ. (29) Γ(β + 1) n=2 Γ(α + β + n) Proof. In view of Deﬁnition 8, it suﬃces to show that z (Qα f (z )) z (Qα f (z )) β β − γ. σ Re +1 (30) (Qα f (z )) (Qα f (z )) β β Then, we have z (Qα f (z )) z (Qα f (z )) z (Qα f (z )) β β β − Re σ (σ + 1) (Qα f (z )) (Qα f (z )) (Qα f (z )) β β β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) (σ + 1)n(n − 1)|an ||z | Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) n−1 1− n=2 Γ(α+β +n) n|an ||z | Γ(β +1) ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) (σ + 1)n(n − 1)an Γ(β +1) . ∞ Γ(α+β +1) Γ(β +n) 1− n=2 Γ(α+β +n) nan Γ(β +1)
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 331 The above expression is bounded by 1 − γ if and only if (29) is satisﬁed. Conversely, if f ∈ UCV QT (α, β, σ, γ ) and z is real, we get ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) 1− ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) nan z Γ(β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) ≥σ + γ. ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n)n an z Γ(β Let z → 1− along the real axis, which gives ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) n(n − 1)an Γ(β +1) (1 − γ ) ≥ (1 + σ ) , ∞ 1 − Γ(α++1) β +1) Γ(β +n) n=2 Γ(α+β +n) nan Γ(β and equivalent to ∞ Γ(α + β + 1) Γ(β + n) (1 − γ ) 1 − nan Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) ≥ (1 + σ ) n(n − 1)an . Γ(β + 1) n=2 Γ(α + β + n) Thus, the proof is complete. Our assertion in Theorem 4 is sharp for functions of the form 1−γ Γ(β + 1) Γ(α + β + n) Fn (z ) = Qα fn (z ) = z − zn, n ≥ 2 β Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )] (31) which belong to the class UCV QT (α, β, σ, γ ). Corollary 7. If f ∈ UCV QT (α, β, σ, γ ) then 1−γ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an (32) Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )] Proof. Since f ∈ UCV QT (α, β, σ, γ ), Theorem 4 gives ∞ Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )]an 1 − γ. Γ(β + 1) n=2 Γ(α + β + n) Next, note that Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )]an Γ(β + 1) Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )]an . Γ(β + 1) n=2 Γ(α + β + n)
332 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim Therefore 1−γ Γ(β + 1) Γ(α + β + n) , n ≥ 2. an Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )] Corollary 8. If f ∈ UCV QT (α, β, σ, γ ) and |z | = r < 1, then (i) |Qα f (z )| r + 2(2+σγ γ ) r2 1− β − (ii) |Qα f (z )| ≥ r − 2(2+σγ γ ) r2 . 1− β − The results are sharp. Proof. First, it is obvious that Γ(α + β + 1) Γ(β + 2) 2(2 + σ − γ )an Γ(β + 1) Γ(α + β + 2) ∞ Γ(α + β + 1) Γ(β + n) n[n(1 + σ ) − (σ + γ )]an , Γ(β + 1) n=2 Γ(α + β + n) and as f ∈ UCV QT (α, β, σ, γ ), using the inequality in Theorem 4 yields ∞ 1−γ Γ(β + 1) Γ(α + β + 2) an . (33) Γ(α + β + 1) Γ(β + 2) 2(2 + σ − γ ) n=2 From (8) with |z | = r(r < 1), we have ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| an r n r+ β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 r+ Γ(β + 1) Γ(α + β + 2) n=2 and ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| ≥ r − an r n β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 . ≥r− Γ(β + 1) Γ(α + β + 2) n=2 Finally, using (33) in the above inequalities gives us both results (i) and (ii). We note that (i) and (ii) are sharp for the following function 1−γ Γ(β + 1) Γ(α + β + 2) Qα f2 (z ) = z − z2 β Γ(α + β + 1) Γ(β + 2) 2(2 + σ − γ ) at z = ±ir, ±r. Deﬁnition 9. Let C QT (α, β, σ ) z (Qα f (z )) z (Qα f (z )) β β f ∈T; +1−σ + 1 + σ , α > 0, β > −1, σ > 0 = Re (Qα f (z )) (Qα f (z )) β β
Uniformly Starlike and Convex Functions Deﬁned by Certain Integral Operator 333 Theorem 5. Let f ∈ T . Then f ∈ C QT (α, β, σ ) if and only if ∞ Γ(α + β + 1) Γ(β + n) n(n − 1 + σ )an σ. (34) Γ(β + 1) n=2 Γ(α + β + n) Proof. By Deﬁnition 9, it is suﬃcient to prove the inequality z (Qα f (z )) z (Qα f (z )) β β −σ 1+ Re +1+σ (Qα f (z )) (Qα f (z )) β β or equivalently z (Qα f (z )) z (Qα f (z )) β β − σ − Re 1 + −σ 1+ 2σ. (Qα f (z )) (Qα f (z )) β β We have z (Qα f (z )) z (Qα f (z )) β β α f (z )) − σ − Re 1 + (Qα f (z )) − σ 1+ (Qβ β z (Qα f (z )) β −σ 2 1+ (Qα f (z )) β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)|an ||z | Γ(β +1) +1−σ 2 ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n|an ||z | Γ(β ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) n(n − 1)an Γ(β +1) +1−σ . 2 ∞ 1 − Γ(α++1) β +1) Γ(β +n) n=2 Γ(α+β +n) nan Γ(β The above expression is bounded by 2σ if and only if (34) is satisﬁed. Conversely, if f ∈ C QT (α, β, σ ) and z is real, we get ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) 1− ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) nan z Γ(β ∞ Γ(α+β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) n(n − 1)an z Γ(β +1) ≥ + 1 − 2σ. ∞ 1 − Γ(α++1) β +1) Γ(β +n) n−1 n=2 Γ(α+β +n) nan z Γ(β Let z → 1− along the real axis, which gives ∞ Γ(α+β +1) Γ(β +n) n=2 Γ(α+β +n) n(n − 1)an Γ(β +1) σ ∞ 1 − Γ(α++1) β +1) Γ(β +n) n=2 Γ(α+β +n) nan Γ(β and gives the required result. Corollary 9. If f ∈ C QT (α, β, σ ) and |z | = r < 1, then for 0 < σ 1 σ (i) |Qα f (z )| r + 2(1+σ) r2 , β σ (i) |Qα f (z )| ≥ r − 2(1+σ) r2 . β Proof. First, it is obvious that
334 Maslina Darus, Aini Janteng, and Suzeini Abdul Halim Γ(α + β + 1) Γ(β + 2) 2(1 + σ )an Γ(β + 1) Γ(α + β + 2) ∞ Γ(α + β + 1) Γ(β + n) n(n − 1 + σ )an , Γ(β + 1) n=2 Γ(α + β + n) and as f ∈ C QT (α, β, σ ), using the inequality in Theorem 5 yields ∞ Γ(β + 1) Γ(α + β + 2) σ an . (35) Γ(α + β + 1) Γ(β + 2) 2(1 + σ ) n=2 From (8) with |z | = r(r < 1), we have ∞ Γ(α + β + 1) Γ(β + n) Qα f (z )| an r n r+ β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 r+ Γ(β + 1) Γ(α + β + 2) n=2 and ∞ Γ(α + β + 1) Γ(β + n) |Qα f (z )| ≥ r − an r n β Γ(β + 1) n=2 Γ(α + β + n) ∞ Γ(α + β + 1) Γ(β + 2) an r 2 . ≥r− Γ(β + 1) Γ(α + β + 2) n=2 Finally, using (35) in the above inequalities gives us both results (i) and (ii). We note that (i) and (ii) are sharp for the following function Γ(β + 1) Γ(α + β + 2) σ Qα f2 (z ) = z − z2 β Γ(α + β + 1) Γ(β + 2) 2(1 + σ ) at z = ±ir, ±r. Remark 2. By taking β = α and α = 1 in (8), analogous results for UCV JT (α, σ ) are also obtained. References 1. R. Bharati, R. Parvatham, and A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math. 28 (1997) 17–33. 2. I. B. Jung, Y. C. Kim, and H. M. Srivastava, The Hardy space of analytic func- tions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993) 138–147. 3. F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993) 189–196. 4. F. Rønning, Integral representations for bounded starlike functions, Annales Polon. Math., 60 (1995) 289–297. 5. H. Silverman, Univalent functions with negative coeﬃcients, Proc. Amer. Math. Soc. 51 (1975) 109–116.