A UNIFIED PRESENTATION OF SOME CLASSES p-VALENT FUNCTIONS WITH FIXED SECOND NEGATIVE COEFFICIENTS

  Bulletin of Mathematics Vol. 03, No. 01 (2011), pp. 61–68.

  

A UNIFIED PRESENTATION OF SOME

CLASSES p-VALENT FUNCTIONS WITH

FIXED SECOND NEGATIVE COEFFICIENTS

_Abstract.

Saibah Siregar

  In this paper we consider the problem involving the unification of some

classes of p- valent functions with fixed second negative coefficients. Coefficient

inequality, growth and distortion theorems are also determined.

  1. INTRODUCTION Let T (p, n) denote the class of functions f of the form: _{X ∞}

  p k

  f a z a p, n (z) = z − k ; k ≥ 0, ∈ N = {1, 2, 3, . . .}, (1)

  k =p+n

  which are analytic and p-valent in the unit disk U = {z : z ∈ C; 0 < |z| < 1}. Let ST α (p, n) and CT α (p, n) be the subclasses of T (p, n) consisting of p- valently starlike functions of order α and p-valently convex functions of order α, respectively, that is, _{( ( ) ) ′} zf

  (z) ST α (p, n) = f ∈ T (p, n) : Re > α (z ∈ U ) f

  (z) Received 10-12-2010, Accepted 15-12-2010.

  2000 Mathematics Subject Classification : 46F20, 52A41 Key words and Phrases : Analytic function, starlike and convex function, unified, growth and distortion

  Saibah Siregar – A Unified Presentation of Some Classes

  and _{( ( ) ) ′′} zf (z) CT f > α ,

  

α (p, n) = ∈ T (p, n) : Re 1 + (z ∈ U )

  f (z) where 0 ≤ α < p. Yamakawa (1992) easily derived the following _{X ∞} f

  ∈ ST α (p, n) ⇐⇒ (k − α)a k ≤ p − α; 0 ≤ α ≤ p, (2)

  k =p+n

  and _{X ∞} f ∈ CT (p, n) ⇐⇒ k (k − α)a ≤ p(p − α); 0 ≤ α ≤ p. (3)

  α k k =p+n

  A function f ∈ T (p, n) is said to be a member of the class T (p, n, α) if it satisfies the inequality: _{( )} f (z) Re > α (z ∈ U ), _′ zf

  (z)

  1 where 0 ≤ α < . p

  We note that T (p, n, α) is a subclass of T (p, n), since _{( ) ( ) ′} f (z) zf (z)

  1 Re > α ⇒ Re > 0; 0 ≤ α < . _′ zf f p (z) (z) For the class T (p, n, α), [3] has given the following Lemma.

  Lemma 1.1 _{X ∞} Let f ∈ T (p, n) satisfies the inequality

  1 (2k − pkα − p)a ≤ p(1 − αp); 0 ≤ α ≤ , (4)

  k

  p

  k =p+n

  then f ∈ T (p, n, α).

  Saibah Siregar – A Unified Presentation of Some Classes

  However, the converse of the lemma is not true and [3] defined the subclass A(p, n, α) of T (p, n, α) consisting of functions f which satisfy (4). And let B(p, n, α) denote the subclass of T (p, n) consisting of functions f such that _′ zf (z) ∈ A(p, n.α).

  Thus Yamakawa [3] gave the following: Lemma 1.2

  A function f defined by (1) is in the class B(p, n, α) if and only if _{X ∞}

  1

  

2

  (2k − pkα − p)a ≤ p (1 − αp); 0 ≤ α ≤ . (5)

  k

  p

  k =p+n

  In view of Lemma 1.1, can be seen that the function f defined by (1) in the class T (p, n, α) _{X ∞} 1 . (2k − pkα − p)a k ≤ p(1 − αp); 0 ≤ α ≤ (6) p

  k =p+n

  so p (1 − αp)

  1 . |a p | ≤ ; 0 ≤ α ≤ (7)

• n

  (p + n)(2 − pα) − p p Let T (p, n, α, c) denote the class of function f (z) in T (p, n, α) of the form _{X ∞} cp (1 − αp) (c − 1)p(1 − αp)

  p p +n p +n+1

  f z z , (z) = z − −

  (p + n)(2 − pα) − p (p + n + 1)(2 − pα) − p

  k =p+n+1

  (8) with 0 < |c| < 1.

  2. PRELIMINARIES To prove our main results, we will need the following definitions and lemmas presented in this section.

  Lemma 2.3 Let the function f be defined by (5) be in the class T (p, n, α, c) and _{X ∞}

  1 . (2k − pkα − p)a k ≤ (1 − c)p(1 − αp); 0 ≤ α ≤ (9) p

  k =p+n with 0 < |c| < 1. Then f (z) ∈ A(p, n, α, c).

  Saibah Siregar – A Unified Presentation of Some Classes Proof.

  By putting p (1 − αp)

  |a | ≤ , (10)

  p +n

  (p + n)(2 − pα) − p with 0 < |c| < 1.

  In (9), will get the results. The result is sharp for function, _{X ∞} cp (1 − αp) (1 − c)p(1 − αp)

  p p +n p +n+1

  f z z , (z) = z − −

  (p + n)(2 − pα) − p (p + n + 1)(2 − pα) − p

  k =p+n+1

  (11) with 0 < |c| < 1.

  Lemma 2.4 Let the function f be defined by (1) be in the class T (p, n, α, c) and _{X ∞}

  1

  2

  k .

  (2k − pkα − p)a k ≤ (1 − c)p (1 − αp); 0 ≤ α ≤ (12) p

  k =p+n with 0 < |c| < 1. Then f (z) ∈ B(p, n, α, c).

  Proof. By putting

  

2

  cp (1 − αp)

  |a p | ≤ ; (13)

• n

  (p + n)(2 − pα) − p with 0 < |c| < 1. The result will have in (12). The results is sharp for function _∞

  2

  cp (1 − αp) (c − 1)p (1 − αp)

  p p +n p +n+1

  f z z , (z) = z − −

  (p + n)(2 − pα) − p (p + n + 1)(2 − pα) − p

  k =p+n+1

  (14) Saibah Siregar – A Unified Presentation of Some Classes

  1 with 0 < |c| < 1 and 0 ≤ α ≤ . p

  In view of (9) and (12), will be introduced and study the some proper- ties and characteristics of the following general class U[p, n, α, λ] of function f ∈ T (p, n) which also satisfy the inequality: _{X ∞} k

  (2k −pkα−p)(1−λ+λ( ))a k ≤ (1−c)p(1−αp); 0 ≤ λ ≤ 1. (15) p

  k =p+n

  It can be see easily U[p, n, α, c, λ] = (1 − λ)A(p, n, c, α) + λB(p, n, α, c), so that

  U[p, n, c, α, 0] = A(p, n, c, α) and U[p, n, c, α, 1] = B(p, n, c, α). The main objective here is to give some properties the unified classes of A(p, n, α) and B(p, n, α) in a more general form U(p, n, α, β).

  The idea is motivated form the work done by Srivastawa [2], and Siregar and Darus [4]. In, Srivastawa the authors gave results on distortion theorem.

  In fact,the properties mentioned for unification of the classes ST (p, n)

  α and CT α (p, n) defined by (2) and (3) respectively can be easily derived.

  Note that when p = 1 and c = 1 in the unification of classes ST (p, n) and

  α

  CT α (p, n).

  3. GROWTH AND DISTORTION THEOREM A growth property for function in the class U [p, n, α, c, λ] is given as following:

  Theorem 3.1 Let the function f defined (1) be in the class U [p, n, c, λ], then cp (1 − αp) (1 − c)p(1 − αp)

  p p +n p +n+1

  |z| − |z| − |z|

  k

  (p + n)(2 − pα) − p (2k − pkα − p)(1 − λ + λ( ))

  p

  cp (1 − αp)

  p p +n

• ≤ |f (z)| ≤ |z| |z|

  (p + n)(2 − pα) − p (1 − c)p(1 − αp)

  p +n+1

  |z| + . (16)

  k

  (2k − pkα − p)(1 − λ + λ( )) Saibah Siregar – A Unified Presentation of Some Classes

  The result is sharp for _{X ∞} cp k

  (1 − αp)

  p p

• n

  f (z) = z − z − (2k − pkα − p)(1 − λ + λ( )a

  k

  p (p + n)(2 − pα) − p

  k =p+n

  ≤ (1 − c)p(1 − αp); (17) with 0 < |c| < 1 and 0 ≤ λ ≤ 1.

  Proof. _{X ∞} (1 − c)p(1 − αp)

  |a k | ≤ ; 0 ≤ λ ≤ 1. (18)

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p k =p+n+1

  Since f ∈ U [p, n, c, λ], then _{X ∞} cp (1 − αp)

  p p k

• n

  |f (z)| ≥ |z| − |z| − |a | |z|

  k

  (p + n)(2 − pα) − p

  k =p+n+1 _∞

  cp ^X (1 − αp)

  p p p

• n +n+1

  ≥ |z| − |z| − |z| |a |

  k

  (p + n)(2 − pα) − p

  k =p+n+1

  cp (1 − αp)

  p p +n

  ≥ |z| − |z| (p + n)(2 − pα) − p

  (1 − c)p(1 − αp)

  p +n+1

  − |z|

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p

  and _{X ∞} cp (1 − αp)

  p p +n k

• |f (z)| ≤ |z| |z| + |a k | |z| (p + n)(2 − pα) − p

  k =p+n+1 _{X ∞}

  cp (1 − αp)

  p p p

• n
• ≤ |z| |z| + |z| |a k |

  (p + n)(2 − pα) − p

  k =p+n+1

  cp (1 − αp)

  p p +n

  ≤ |z| |z| + (p + n)(2 − pα) − p

  (1 − c)p(1 − αp)

  p

• n+1 |z| + .

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p The proof is complete.

  The distortion property for function in the class U [p, n, α, c, λ] is given as following:

  Saibah Siregar – A Unified Presentation of Some Classes

  Theorem 3.2 Let the function f defined (1) be in the class U [p, n, c, λ], then cp (p + n)(1 − αp) (1 − c)p(p + n + 1)(1 − αp)

  p− 1 p +n−1 p +n

  p |z| − |z| − |z|

  k

  (p + n)(2 − pα) − p (2k − pkα − p)(1 − λ + λ( ))

  p _{′ (p + n)(1 − αp)} cp

p− p

1 +n−1

  ≤ |f + (z)| ≤ p|z| |z| (p + n)(2 − pα) − p

  (1 − c)p(p + n + 1)(1 − αp)

  p

|z| (19)

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p

  The result is sharp for _{X ∞} cp (1 − αp) k

  p p +n

  f z (z) = z − − (2k − pkα − p)(1 − λ + λ( )a k

  (p + n)(2 − pα) − p p

  k =p+n

  ≤ (1 − c)p(1 − αp); (20) with 0 < |c| < 1 and 0 ≤ λ ≤ 1.

  Proof. _{X ∞} (1 − c)p(1 − αp)

  |a k | ≤ ; 0 ≤ λ ≤ 1. (21)

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p k =p+n+1

  Since f ∈ U [p, n, c, λ], then _∞ cp ^X _′ (p + n)(1 − αp)

  1

  p− 1 p +n−1 k−

  |f (z)| ≥ p|z| − |z| − k |a | |z|

  k

  (p + n)(2 − pα) − p

  k =p+n+1 _{X ∞}

  cp (p + n)(1 − αp)

  p−

1 p +n−1 p +n

  k ≥ p|z| − |z| − |z| |a k |

  (p + n)(2 − pα) − p

  k =p+n+1

  cp (p + n)(1 − αp)

  p− p 1 +n−1

  ≥ p|z| − |z| (p + n)(2 − pα) − p

  (1 − c)p(p + n + 1)(1 − αp)

  p

• n

  − |z|

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p Saibah Siregar – A Unified Presentation of Some Classes

  and _{X ∞ ′} cp (p + n)(1 − αp)

  1

  p− 1 p +n−1 k−

  |f (z)| ≤ p|z| + |z| |a k | |z| (p + n)(2 − pα) − p

• k

  k =p+n+1 _{X ∞}

  cp (p + n)(1 − αp)

  

p− p p

1 +n−1 +n

  ≤ p|z| |z| + |z| |a k | (p + n)(2 − pα) − p

• k

  k =p+n+1

  cp (p + n)(1 − αp)

  p− 1 p +n−1

• ≤ p|z| |z|

  (p + n)(2 − pα) − p (1 − c)p(p + n + 1)(1 − αp)

  p

• n

  |z| +

  k

  (2k − pkα − p)(1 − λ + λ( ))

  p The proof is complete.

  References

  [1] Siregar, S. & Darus, M. Unified treatment of p-valently analytic func- tions. Far East J. Math. Sci. 2005,17(1):69–79 [2] Srivastawa, H. M., Owa, S., Obradovic, M. and Nikic,M., A unified presentation of certain classes of starlike and convex functions with negative coefficients, Utilitas Math. 36(1989), 107–113. [3] Yamakawa, R., Certain subclasses of p-valently starlike functions with negative coefficients. in H. M. Srivastava & S. Owa, (eds.), Current topics in analytic function theory, Publishing Company, New Jersey, 1992, 393-402. [4] Siregar, S. and Darus, M. A unified presentation of some classes of meromorphically p-valent functions with fixed second negative coeffi- cients. Proc. Simposium Kebangsaan Sains Matematik ke-12 by IIUM, 2004, pg. 116-130.

  Saibah Siregar : Faculty of Science and Biotechnology, Universiti Industri Selan- gor, 43600 Bestari Jaya, Selangor D.E., Malaysia

  E-mail: saibahmath@yahoo.com