International Journal of Pure and Applied Mathematics

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 62 No. 3 2010, 337-346

ON DIFFERENTIAL OPERATOR FOR MULTIVALENT FUNCTIONS Maslina Darus1 § , Rabha W. Ibrahim2 1,2 School

of Mathematical Sciences Faculty of Science and Technology University Kebangsaan Malaysia Selangor Darul Ehsan, Bangi, 43600, MALAYSIA 1 e-mail: [email protected] 2 e-mail: [email protected] Abstract: In this article we define a differential operator for multivalent functions in an open disk. Further, we define some classes containing this operator. Partial sums are considered. AMS Subject Classification: 30C45 Key Words: differential operator, multivalent functions, partial sums 1. Introduction Let T (p) denote the class of functions f of the form ∞ X p an z n+p (p ∈ N, z ∈ U ), f (z) = z +

(1)

n=1

which are analytic and p-valent (multivalent) in the open unit disk U = {z : z ∈ C and |z| < 1}. Let bePgiven two functions f, g ∈ T (p), f (z) = z p + P ∞ n+p and g(z) = z p + ∞ n convolution or Hadamard n=1 an z n=p+1 bn z . Then their P∞ p product f (z) ∗ g(z) is defined by f (z) ∗ g(z) = z + n=1 an bn z n+p (z ∈ U ). Define a function ϕp (a, c; z) as follows ∞ X (a)n n+p p z , c 6= 0, −1, −2, ..., ϕp (a, c; z) := z + (c)n n=1 where (a)n is the Pochhammer symbol. Received:

June 23, 2010

§ Correspondence

author

c 2010 Academic Publications

338

M. Darus, R.W. Ibrahim Assuming that a = k + p > 0 and c = 1 for k = 0, 1, 2, ... in ϕp (a, c; z) yields ∞ X (k + p)n n+p p z . (2) ϕp (k + p, 1; z) = z + (1)n n=1

Next we define the following differential operator Dλk : T (p) → T (p) by ∞ X an z n+p , D0 f (z) = f (z) = z p + n=1

1 Dλ,p f (z) = (1 − λp)f (z) + λzf ′ (z) = z p +

∞ X

(1 + λn)an z n+p , (3)

n=1

.. . k f (z) = z p + Dλ,p

∞ X

(1 + λn)k an z n+p

(z ∈ U ),

n=1

where

p ∈ N, k ∈ N0 , λ ≥ 0 .

Again by applying convolution product on (2) and (3) we have the following operator zp k k f (z) = Dλ,p f (z) ∗ Dλ,p (1 − z)k+p ∞ X (k + p)n = zp + (1 + λn)k an z n+p (4) (1)n = zp +

n=1 ∞ X

C(n, k)(1 + λn)k an z n+p

(z ∈ U ),

n=1

where C(n, k) :=

(k+p)n (1)n .

k f (z), when λ = 0, p = 1, was introduced by Remark 1. The symbol Dλ,p Ruscheweyh [2] and when λ = 0, see Goel and Sohi [1].

f ∈ T (p) is said to be p-valent starlike of order µ, 0 ≤ µ µ (z ∈ U ), this class is denoted by Sp∗ (µ). A function f ∈ T (p) is n o ′′ (z) said to be p-valent convex of order µ, 0 ≤ µ µ (z ∈ U ), this class is denoted by Cp (µ). A function f ∈ T (p) is said to be in the class Sp∗ (µ, λ) of order µ, where n z[Dk f (z)]′ o 0 ≤ µ µ (z ∈ U ). A function f ∈ T (p) is said to be λ,p

ON DIFFERENTIAL OPERATOR FOR... n in the class Cp (µ, λ) of order µ, where 0 ≤ µ µ

(z ∈ U ). For 0 ≤ α β − p (5) (z ∈ U ). k k Dλ,p f (z) Dλ,p f (z) Also, for 0 ≤ α β − (p − 1) (z ∈ U ). (6) ℜ 1+ k k ′ ′ [Dλ,p f (z)] [Dλ,p f (z)]

The main aim of this work is to study coefficient bounds and extreme points of the general classes Sp∗ (α, β, λ) and Cp (α, β, λ). Partial sums fm of functions f in the classes Sp∗ (α, β, λ) and Cp (α, β, λ) are considered.

2. The Classes Sp∗(α, β, λ) and Cp(α, β, λ) In this section we obtain a sufficient condition and extreme points for functions f in the classes Sp∗ (α, β, λ) and Cp (α, β, λ). Theorem 1. A sufficient condition for a function f of the form (1) to be in Sp∗ (α, β, λ), 0 ≤ α < p, β ≥ 0 is ∞ X [(1 + β)n + (p − α)]C(n, k)(1 + λn)k |an | < p − α (z ∈ U ). (7) n=1

Proof. It suffices to show that z[D k f (z)]′ n z[D k f (z)]′ o λ,p λ,p − p − ℜ − p ≤ p−α β k f (z) k f (z) Dλ,p Dλ,p

(z ∈ U ).

z[D k f (z)]′ z[D k f (z)]′ n z[D k f (z)]′ o λ,p λ,p λ,p β − ℜ − p − p ≤ (1 + β) − p k k k Dλ,p f (z) Dλ,p f (z) Dλ,p f (z) P (1 + β) ∞ nC(n, k)(1 + λn)k |an ||z|n+p P∞n=1 ≤ 1 − n=1 C(n, k)(1 + λn)k |an ||z|n+p P nC(n, k)(1 + λn)k |an | (1 + β) ∞ P∞ n=1 . ≤ 1 − n=p+1 C(n, k)(1 + λn)k |an |

340

M. Darus, R.W. Ibrahim

This last expression is bounded above by (p − α) if ∞ X [(1 + β)n + (p − α)]C(n, k)(1 + λn)k |an | < p − α, n=1

and the proof is complete. In the same manner we can obtain the next result. Theorem 2. A sufficient condition for a function f of the form (1) to be in Cp (α, β, λ), 0 ≤ α < p, β ≥ 0 is ∞ X (n + p)[n(1 + β) + (p − α)]C(n, k)(1 + λn)k |an | < p(p − α). (8) n=1

Proof. It suffices to show that for p ∈ N 0 ≤ α < p, β ≥ 0, z ∈ U z[D k f (z)]′′ o n z[D k f (z)]′′ λ,p λ,p β − (p − 1) − (p − 1) ≤ p − α, − ℜ k f (z)]′ k f (z)]′ [Dλ,p [Dλ,p z[D k f (z)]′′ n z[D k f (z)]′′ o λ,p λ,p β − (p − 1) − ℜ − (p − 1) k f (z)]′ k f (z)]′ [Dλ,p [Dλ,p z[D k f (z)]′′ λ,p − (p − 1) ≤ (1 + β) k ′ [Dλ,p f (z)] P k n+p−1 (1 + β) ∞ n=1 n(n + p)C(n, k)(1 + λn) |an ||z| P ≤ p|z|p−1 − ∞ (n + p)C(n, k)(1 + λn)k |an ||z|n+p−1 P∞ n=1 (1 + β) n=1 n(n + p)C(n, k)(1 + λn)k |an | P ≤ . k p− ∞ n=1 (n + p)C(n, k)(1 + λn) |an | This last expression is bounded above by (p − α) if ∞ X (n + p)[n(1 + β) + (p − α)]C(n, k)(1 + λn)k |an | < p(p − α) (p ∈ N). n=1

This completes the proof.

3. Partial Sums In this section, applying methods used by Silverman [3] and Silvia [4], we will investigate the ratio of a function defined in (1) to its sequence of partial sums m X p an z n+p (z ∈ U ) (9) fm+p (z) = z + n=1

ON DIFFERENTIAL OPERATOR FOR...

341

when the coefficients are small enough in order to satisfy either condition (7) or (8). More precisely, we will determine sharp lower bounds for o n f ′ (z) o n f ′ (z) o n f (z) o n f m+p m+p (z) ,ℜ ,ℜ ′ and ℜ . ℜ fm+p (z) f (z) fm+p (z) f ′ (z) n o In the sequel, we will make use of the fact that ℜ (1+w(z)) (1−w(z)) > 0 (z ∈ U ) if and P∞ n only if w(z) = n=1 cn z satisfies the inequality |w(z)| < |z|. Theorem 3. Let f given by (1) satisfy (7). Then n f (z) o >1 ℜ fm+p (z) p−α , (10) − [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k where z ∈ U , p > α, m = 0, 1, 2, ....

The result is sharp for every m with the extremal function p−α f (z) = z p + [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k × z m+p+1 . (11) Proof. Assume that f ∈ T (p) satisfies (7). By setting [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k p−α n f (z) × fm+p (z) o p−α − 1− [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k P Hm+p+1 ∞ an z n Pmn=m+1 n := 1 + , 1 + n=1 an z where w(z) =

Hm+p+1 [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k . p−α Thus we find that P w(z) − 1 Hm+p+1 ∞ n=m+1 |an | Pm P ≤ 1 (z ∈ U ) ≤ w(z) + 1 2 − 2 n=1 |an | − Hm+p+1 ∞ n=m+1 |an | :=

342


if and only if 2Hm+p+1

∞ X

|an | ≤ 2 − 2

m X

|an |

n=1

n=m+1

which is equivalent to m X

|an | + Hm+p+1

n=1

∞ X

|an | ≤ 1.

(12)

n=m+1

In order to see that f (z) = z p +

z m+p+1 Hm+p+1

(z ∈ U ) πi

gives a sharp result, we observe that for z = re m+p (z ∈ U ) that z m+p 1 f (z) =1+ →1− as z → 1− . fm+p (z) Hm+p+1 Hm+p+1 This completes the proof. Theorem 4. Let f given by (1) satisfy (7). Then o nf m+p (z) > ℜ f (z) [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k , (p − α) + [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (13) where z ∈ U , p > α, m = 0, 1, 2, .... The result is sharp for every m with an extremal function given by (11). Proof. Assume that f ∈ T (p) and satisfies (7). Write w(z) = [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k 1+ p−α nf m+p (z) × f (z) o [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k − (p − α) + [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k P ∞ n 1 + Hm+p+1 n=m+1 an z P =1− , n 1+ m n=1 an z


343

where Hm+p+1 is defined in Theorem 3 This yields that P w(z) − 1 (1 + Hm+p+1 ) ∞ n=m+1 |an | Pm P ≤ 1 (z ∈ U ) ≤ w(z) + 1 2 − 2 n=p+1 |an | − (1 + Hm+p+1 ) ∞ n=m+1 |an | if and only if

2[(1 + Hm+p+1 )

∞ X

|an |] ≤ 2 − 2

m X

|an |

n=2

n=m+1

or m X

|an | + (1 + Hm+p+1 )

∞ X

|an | ≤ 1,

(14)

n=m+1

n=p+1

which gives (13). The bound in (13) is sharp for all m ∈ N with the extremal function given by (11). This completes the proof. Theorem 5. Let f given by (1) satisfy (7). Then n f ′ (z) o ℜ ′ ≥1 fm+p (z) (m + p + 1)(p − α) , (15) [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k where z ∈ U , p > α, m = 0, 1, 2, .... −

Proof. Assume that f ∈ T (p) satisfies (7). Write w(z) =

− 1−

[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k p−α n f ′ (z) × ′ fm+p (z) o (m + p + 1)(p − α)

[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k P∞ n+p Hm+p+1 P∞ n+p n n 1 + (m+p+1) n=1 p an z n=m+1 p an z + = Pm n+p 1 + n=1 p an z n Hm+p+1 P∞ n+p n n=m+1 p an z (m+p+1) =1+ , P n+p n 1+ m n=1 p an z

where Hm+p+1 is defined in Theorem 3. This implies Hm+p+1 P∞ n+p w(z) − 1 n=m+1 p |an | m+p+1 ≤ P P H w(z) + 1 2 − 2 m n+p |an | − m+p+1 ∞ n=1

p

m+p+1

n+p n=m+1 p |an |

≤ 1 (z ∈ U )

344


if and only if m ∞ X n Hm+p+1 X n |an |] ≤ 2 − 2 |an |, 2[ m+p+1 p p n=p+1

n=m+1

i.e. m+p X n=1

∞ Hm+p+1 X n n |an | + |an | ≤ 1. p m+p+1 p n=m+1

We therefore obtain (15). The result is sharp with functions given by (11). The proof of Theorem 5 is completed. Theorem 6. Let f given by (1) satisfy (7). Then n f ′ (z) o ≥ ℜ m′ f (z) [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k . (m + p + 1)(p − α) + [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (16)

Proof. Assume that f ∈ T (p) satisfies (7). Consider “ [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k ” w(z) = (m + p + 1) + p−α n f ′ (z) m × f ′ (z) o [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k − k (m + p + 1)(p − α) + [(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1)) Hm+p+1 P∞ (1 + m+p+1 ) n=m+1 n+p an z n p . =1− Pm n+p 1 + n=2 p an z n

This implies that Hm+p+1 P∞ w(z) − 1 ) n=m+1 n+p (1 + m+p+1 p |an | ≤ Pm n+p P H m+p+1 w(z) + 1 2 − 2 n=1 p |an | − (1 + m+p+1 ) ∞ n=m+1

n+p p |an |

if and only if 2[(1 +

∞ m X X Hm+p+1 n+p n+p ) |an |] ≤ 2 − 2 |an |, m+p+1 p p n=m+1

n=1

i.e. m X n+p

n=1

p

∞ X Hm+p+1 n+p |an | + (1 + ) |an | ≤ 1. m + p + 1 n=m+1 p

≤1 (z ∈ U )


345

We therefore obtain (16). The result is sharp with functions given by (11). The proof of Theorem 6 is complete. In the same manner as the proof of Theorems 3-6, we can show the following results: Theorem 7. Let f given by (1) satisfy (8). Then n f (z) o >1 ℜ fm+p (z) p(p − α) − (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (z ∈ U ). (17) The result is sharp for every m with the extremal function f (z) = z p + p(p − α) (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k × z m+p+1 . (18) Theorem 8. Let f given by (1) satisfy (8). Then nf o m+p (z) ℜ > f (z) (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k . p(p − α) + (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (19)

The result is sharp for every m with the extremal function given by (18). Theorem 9. Let f given by (1) satisfy (8). Then n f ′ (z) o ℜ ′ ≥ 1− fm+p (z) p(m + p + 1)(p − α) . (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (20) Theorem 10. Let f given by (1) satisfy (8). Then ℜ

n f ′ (z) o m ≥ f ′ (z)

(m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k . p(m + p + 1)(p − α) + (m + p + 1)[(1 + β)(m + p + 1) + (p − α)]C(m + p + 1, k)(1 + λ(m + p + 1))k (21)

346

M. Darus, R.W. Ibrahim Acknowledgments

The work here was supported by UKM-ST-06-FRGS0107-2009, MOHE Malaysia.

References [1] R.M. Goel, N.S. Sohi, A new criterion for p-valent functions, Proc. Amer. Math. Soc., 78 (1980), 353-357. [2] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 49 (1975), 109-115. [3] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl., 209 (1997), 221-227. [4] E.M. Silvia, Partial sums of convex functions of order α, Houston J. Math., 11, No. 3 (1985), 397-404.