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Volume 10 (2009), Issue 4, Article 112, 6 pp. ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT FUNCTIONS INVOLVING THE GENERALIZED SAITOH OPERATOR HESAM MAHZOON AND S. LATHA D EPARTMENT OF S TUDIES IN M ATHEMATICS M ANASAGANGOTRI U NIVERSITY OF M YSORE - INDIA. [email protected] D EPARTMENT OF M ATHEMATICS Y UVARAJA’ S C OLLEGE U NIVERSITY OF M YSORE -INDIA. [email protected] Received 10 June, 2009; accepted 03 November, 2009 Communicated by S.S. Dragomir A BSTRACT. In this paper, we introduce the generalized Saitoh operator Lp (a, c, η) and using p,b p,b,α this operator, the new subclasses Hn,m (a, c, η), Lp,b and n,m (a, c, η; µ), Hn,m (a, c, η) p,b,α Ln,m (a, c, η; µ) of the class of multivalent functions denoted by Ap (n) are defined. Further for functions belonging to these classes, certain properties of neighborhoods are studied. Key words and phrases: Coefficient bounds, (n, δ) - neighborhood and generalized Saitoh operator. 2000 Mathematics Subject Classification. 30C45. 1. I NTRODUCTION Let Ap (n) be the class of normalized functions f of the form (1.1) ∞ X p f (z) = z + ak z k , (n, p ∈ N), k=n+p which are analytic and p-valent in the open unit disc U = {z ∈ C : |z| < 1}. Let Tp (n) be the subclass of Ap (n), consisting of functions f of the form (1.2) p f (z) = z − ∞ X ak z k , (ak ≥ 0, n, p ∈ N), k=n+p which are p-valent in U. The Hadamard product of two power series p f (z) = z + ∞ X k=n+p 155-09 ak z k and p g(z) = z + ∞ X k=n+p bk z k 2 H ESAM M AHZOON AND S. L ATHA is defined as (f ∗ g)(z) = z p + ∞ X ak b k z k . k=n+p Z− 0, Definition 1.1. For a ∈ R, c ∈ R \ where Z− 0 = {..., −2, −1, 0} and η ∈ R (η ≥ 0), the operator Lp (a, c, η) : Ap (n) → Ap (n), is defined as Lp (a, c, η)f (z) = φp (a, c, z) ∗ Dη f (z), (1.3) where η Dη f (z) = (1 − η)f (z) + zf 0 (z), p and (η ≥ 0, z ∈ U) ∞ X (a)k−p k φp (a, c, z) = z + z , (c)k−p k=n+p p z∈U and (x)k denotes the Pochammer symbol given by ( 1 if k = 0, (x)k = x(x + 1) · · · (x + k − 1) if k ∈ N = {1, 2, 3, ...}. In particular, we have, LP 1 (a, c, η) ≡ L(a, c, η). k Further, if f (z) = z p + ∞ k=n+p ak z , then ∞ X (a)k−p k p Lp (a, c, η)f (z) = z + −1 η ak z k . 1+ p (c) k−p k=n+p Remark 1. For η = 0 and n = 1, we obtain the Saitoh operator [7] which yields the Carlson Shaffer operator [1] for η = 0 and n = p = 1. For any function f ∈ Tp (n) and δ ≥ 0, the (n, δ)-neighborhood of f is defined as, ( ) ∞ ∞ X X (1.4) Nn,δ (f ) = g ∈ Tp (n) : g(z) = z p − bk z k and k|ak − bk | ≤ δ . k=n+p k=n+p p For the function h(z) = z , (p ∈ N) we have, ( (1.5) Nn,δ (h) = g ∈ Tp (n) : g(z) = z p − ∞ X k=n+p bk z k and ∞ X ) k|bk | ≤ δ . k=n+p The concept of neighborhoods was first introduced by Goodman [2] and then generalized by Ruscheweyh [6] . p,b Definition 1.2. A function f ∈ Tp (n) is said to be in the class Hn,m (a, c, η) if ! 1 z (L (a, c, η)f (z))(m+1) p (1.6) − (p − m) < 1, (m) b (Lp (a, c, η)f (z)) where p ∈ N, m ∈ N0 , a > 0, η ≥ 0, p > m, b ∈ C \ {0} and z ∈ U. Definition 1.3. A function f ∈ Tp (n) is said to be in the class Lp,b n,m (a, c, η; µ) if " # (m) 1 L (a, c, η)f (z) p (m+1) (1.7) p(1 − µ) + µ (Lp (a, c, η)f (z)) − (p − m) < p−m, b z where p ∈ N, m ∈ N0 , a > 0, η ≥ 0, p > m, µ ≥ 0, b ∈ C \ {0} and z ∈ U. J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp. http://jipam.vu.edu.au/ G ENERALIZED S AITOH O PERATOR 3 2. C OEFFICIENT B OUNDS In this section, we determine the coefficient inequalities for functions to be in the subclasses and Lp,b n,m (a, c, η; µ). p,b Hn,m (a, c, η) p,b Theorem 2.1. Let f ∈ Tp (n). Then, f ∈ Hn,m (a, c, η) if and only if ∞ X (a)k−p k p k . 1+ −1 η (k + |b| − p) ak ≤ |b| m p (c) k−p m k=n+p (2.1) p,b Proof. Let f ∈ Hn,m (a, c, η). Then, by (1.6) and (1.7) we can write, h i P (a)k−p k k k−p ∞ (p − k)a z 1 + − 1 η k k=n+p p (c)k−p m h i (2.2) < > −|b|, P (a)k−p k k p − ∞ k−p 1 + − 1 η a z k=n+p m p (c)k−p m k (z ∈ U). Taking z = r, (0 ≤ r < 1) in (2.2), we see that the expression in the denominator on the left hand side of (2.2), is positive for r = 0 and for all r, 0 ≤ r < 1. Hence, by letting r 7→ 1− through real values, expression (2.2) yields the desired assertion (2.1). Conversely, by applying the hypothesis (2.1) and letting |z| = 1, we obtain, z (L (a, c, η)f (z))(m+1) p − (p − m) (m) (Lp (a, c, η)f (z)) h i P (a)k−p k ∞ k k−m 1 + − 1 η (p − k)a z k k=n+p p (c)k−p m h i = P (a)k−p k k k−m mp z p−m − ∞ 1 + − 1 η a z k=n+p p (c)k−p m k h P h i i (a)k−p k k |b| mp − ∞ 1 + − 1 η a k k=n+p p (c)k−p m h i ≤ P (a)k−p k p k − ∞ k=n+p 1 + p − 1 η (c)k−p m ak m = |b|. p,b Hence, by the maximum modulus theorem, we have f ∈ Hn,m (a, c, η). On similar lines, we can prove the following theorem. Theorem 2.2. A function f ∈ Lp,b n,m (a, c, η; µ) if and only if ∞ X k (a)k−p k − 1 (2.3) 1+ −1 η [µ(k − 1) + 1] ak p (c) m k−p k=n+p |b| − 1 p ≤ (p − m) + . m! m 3. I NCLUSION R ELATIONSHIPS I NVOLVING (n, δ)-N EIGHBORHOODS In this section, we prove certain inclusion relationships for functions belonging to the classes and Lp,b n,m (a, c, η; µ). p,b Hn,m (a, c, η) Theorem 3.1. If (3.1) (n + p)|b| mp δ= n n (n + |b|) 1 + p η (a) (c)n J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp. , n+p m (p > |b|), http://jipam.vu.edu.au/ 4 H ESAM M AHZOON AND S. L ATHA p,b then Hn,m (a, c, η) ⊂ Nn,δ (h). p,b Proof. Let f ∈ Hn,m (a, c, η). By Theorem 2.1, we have, ∞ (a)n n + p X p n ak ≤ |b| , (n + |b|) 1 + η m p (c)n m k=n+p which implies ∞ X (3.2) k=n+p ak ≤ |b| p m n (n + |b|) 1 + np η (a) (c)n . n+p m Using (2.1) and (3.2), we have, ∞ n (a)n n + p X 1+ η kak p (c)n m k=n+p ∞ p n (a)n n + p X ≤ |b| + (p − |b|) 1 + η ak m p (c)n m k=n+p |b| mp p n (a)n n + p ≤ |b| + (p − |b|) 1 + η n m p (c)n m (n + |b|) 1 + np η (a) (c)n p n+p . = |b| m n + |b| That is, ∞ X |b|(n + p) mp kak ≤ n (n + |b|) 1 + np η (a) k=n+p (c)n = δ, n+p m n+p m (p > |b|). Thus, by the definition given by (1.5), f ∈ Nn,δ (h). Similarly, we prove the following theorem. Theorem 3.2. If (3.3) h i p (p − m)(n + p) |b|−1 + m! m δ= , n n+p [µ(n + p − 1) + 1] 1 + np η (a) (c)n m (µ > 1) then Lp,b n,m (a, c, η; µ) ⊂ Nn,δ (h). 4. F URTHER N EIGHBORHOOD P ROPERTIES In this section, we determine the neighborhood properties of functions belonging to the subp,b,α classes Hn,m (a, c, η) and Lp,b,α n,m (a, c, η; µ). p,b,α For 0 ≤ α < p and z ∈ U, a function f is said to be in the class Hn,m (a, c, η) if there p,b exists a function g ∈ Hn,m (a, c, η) such that f (z) < p − α. (4.1) − 1 g(z) For 0 ≤ α < p and z ∈ U, a function f is said to be in the class Lp,b,α n,m (a, c, η; µ) if there p,b exists a function g ∈ Ln,m (a, c, η; µ) such that the inequality (4.1) holds true. J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 112, 6 pp. http://jipam.vu.edu.au/ G ENERALIZED S AITOH O PERATOR 5 p,b Theorem 4.1. If g ∈ Hn,m (a, c, η) and (4.2) n n+p δ(n + |b|) 1 + np η (a) (c)n m h α=p− n n n+p (n + p) (n + |b|) 1 + p η (a) − |b| (c)n m i , p m p,b,α then Nn,δ (g) ⊂ Hn,m (a, c, η). Proof. Let f ∈ Nn,δ (g). Then, ∞ X (4.3) k|ak − bk | ≤ δ, k=n+p which yields the coefficient inequality, ∞ X (4.4) |ak − bk | ≤ k=n+p δ , n+p (n ∈ N). p,b Since g ∈ Hn,m (a, c, η), by (3.2) we have, (4.5) ∞ X bk ≤ k=n+p |b| p m n (n + |b|) 1 + np η (a) (c)n n+p m so that, P∞ f (z) |ak − bk | < k=n+p P − 1 ∞ g(z) 1 − k=n+p bk (a)n n+p n (n + |b|) 1 + η p (c)n m δ h ≤ n + p (n + |b|) 1 + n η (a)n n+p − |b| p (c)n m i p m = p − α. p,b,α Thus, by definition, f ∈ Hn,m (a, c, η) for α given by (4.2). On similar lines, we prove the following theorem. Theorem 4.2. If g ∈ Lp,b n,m (a, c, η; µ) and (4.6) n n+p−1 δ[µ(n + p − 1) + 1] 1 + np η (a) (c)n m h α=p− (a)n n+p−1 n + (n + p) {µ(n + p − 1) + 1} 1 + p η (c)n m − (p − m) |b|−1 m! i , p m then Nn,δ (g) ⊂ Lp,b,α n,m (a, c, η; µ). R EFERENCES [1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM. J. Math. Anal.,15(4) (1984), 737–745. [2] A.W. GOODMAN, Univalent functions and non-analytic curves, Proc. Amer. Math. 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