On Certain Multivalent Functions

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2007, Article ID 72393, 5 pagesdoi:10.1155/2007/72393

Research ArticleOn Certain Multivalent Functions

Mamoru Nunokawa, Shigeyoshi Owa, Tadayuki Sekine, Rikuo Yamakawa,Hitoshi Saitoh, and Junichi Nishiwaki

Received 11 February 2007; Accepted 17 April 2007

Recommended by Linda R. Sons

Let �∗(p,α) be the class of functions f (z) which are analytic and p-valently starlike oforder α in the open unit disk E. The object of the present paper is to derive an interestingcondition for f (z) to be in the class �∗(p,α).

Copyright © 2007 Mamoru Nunokawa et al. This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distribu-tion, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let �(p) denote the class of functions f (z) of the form

f (z)= zp +∞∑

n=p+1

anzn(p ∈N= {1,2,3, . . .}) (1.1)

which are analytic in the open unit disk E= {z : z ∈ C, |z| < 1}. A function f (z)∈�(p)is said to be p-valently starlike of order α(0 � α < p) in E if and only if

Rez f ′(z)f (z)

> α (z ∈ E). (1.2)

We denote by �∗(p,α) the subclass of �(p) consisting of functions which are p-valentlystarlike of order α in E. We only call a function f (z)∈�∗(p,0) to be p-valently starlike inE. Further, a function f (z)∈�(p) is said to be p-valently convex of order α(0 � α < p)in E if and only if

1 + Rez f ′′(z)f ′(z)

> α (z ∈ E). (1.3)

2 International Journal of Mathematics and Mathematical Sciences

We denote by �(p,α) the subclass of �(p) consisting of all p-valently convex functions oforder α in E. We also call a function f (z)∈�(p,0) p-valently convex function. From thedefinition, it is trivial that if f (z) is a p-valently convex function, then z f ′(z) is p-valentlystarlike in E.

2. Preliminaries

In this paper, we need the following lemmas.

Lemma 2.1. If M(z) = zp +∑∞

n=p+k anzn (1 � p and 1 � k) and N(z) = zp +

∑∞n=p+1 anz

n

are analytic in E and N(z) satisfies Re(N(z)/zN ′(z)) > δ (0 � δ < 1/p), then

Re(M′(z)N ′(z)

)> β implies Re

(M(z)N(z)

)>

2β+ kδ

2 + kδ. (2.1)

Remark 2.2. This lemma holds to be true for N(z) which is multivalently starlike in E.

We owe the above lemma to Ponnusamy and Karunakaran [1].

Lemma 2.3. Let f (z)∈�∗(p,0). Then

F(z)p+ 1

=∫ z

0f (t)dt ∈ S∗(p+ 1,0) (2.2)

or

RezF′(z)F(z)

> 0 (z ∈ E). (2.3)

The proof of this result can be found in [2].

3. Main result

Theorem 3.1. If f (z)∈�(p) satisfies the following condition:

Rez f (p)(z)f (p−1)(z)

> α (z ∈ E) (3.1)

for α(0 � α < 1), then f (z)∈�∗(p,α+ p− 1) or

Rez f ′(z)f (z)

> α+ p− 1 (z ∈ E). (3.2)

Proof. From assumption (3.1), f (p−1)(z) is univalently starlike of order α and it is trivialthat

(z f (p−1)(z)− f (p−2)(z)

)′ = z f (p)(z). (3.3)

Note that f (p−2)(z) is starlike in E by Lemma 2.3. Therefore, applying Lemma 2.1 and(3.3), we have

Rez f (p−1)(z)− f (p−2)(z)

f (p−2)(z)> α (z ∈ E). (3.4)

Mamoru Nunokawa et al. 3

From Lemma 2.3, f (p−2)(z) is 2-valently starlike in E. Now then, it is trivial that

(z f (p−2)(z)− 2 f (p−3)(z)

)′ = z f (p−1)(z)− f (p−2)(z). (3.5)

Then, from Lemma 2.1, 2-valently starlikeness of f (p−2)(z), and (3.5), we have

Rez f (p−2)(z)− 2 f (p−3)(z)

f (p−3)(z)> α (z ∈ E). (3.6)

Further, it is trivial that

(z f (p−3)(z)− 3 f (p−4)(z)

)′ = z f (p−2)(z)− 2 f (p−3)(z), (3.7)

and applying the same method and reason as above, we have

Rez f (p−3)(z)− 3 f (p−4)(z)

f (p−4)(z)> α (z ∈ E), (3.8)

where f (p−3)(z) is 3-valently starlike in E. Applying the mathematical induction, we have

Rez f ′(z)− (p− 1) f (z)

f (z)> α (z ∈ E), (3.9)

where f (z) is p-valently starlike in E. This shows that

Rez f ′(z)f (z)

> α+ p− 1 (z ∈ E), (3.10)

or f (z)∈�∗(p,α+ p− 1). �

Our main result shows the following.

Corollary 3.2. Let f (z)∈�(p), 2 � p, 0 � α < 1,

Rez f (p)(z)f (p−1)(z)

> α (z ∈ E), (3.11)

and put f (z)= zp−1 f1(z) where

f1(z)= z+∞∑

n=p+1

anzn−p+1. (3.12)

Then f1(z) is univalently starlike of order α in E.

Proof. From the definition of f1(z) and Theorem 3.1, it follows that

Rez f ′(z)f (z)

= p− 1 + Rez f ′1 (z)f1(z)

> α+ p− 1. (3.13)

This completes the proof. �

4 International Journal of Mathematics and Mathematical Sciences

Corollary 3.3. Let f (z)∈�(p), 2 � p, 0 � α < 1, and

1 + Rez f (p+1)(z)f (p)(z)

> α (z ∈ E). (3.14)

Then one has

Rez f ′(z)f (z)

> β(α) + p− 1 (z ∈ E), (3.15)

where

β(α)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1− 2α22−2α

[1− 22α−1

] if α �= 12

,

12log2

if α= 12.

(3.16)

Proof. Putting

g(z)= f (p−1)(z)p!

= z+∞∑

n=2

bnzn, (3.17)

then from assumption (3.14), g(z) is univalently convex of order α, and therefore fromWilken-Feng result [3] and Theorem 3.1, we have

Rezg′(z)g(z)

= Rez f (p)(z)f (p−1)(z)

> β(α) (z ∈ E), (3.18)

and therefore it follows that

Rez f ′(z)f (z)

> β(α) + p− 1 (z ∈ E). (3.19)

�

We will give here an open problem.

Problem 3.4. Let f (z)∈�∗(p,α) and 0 � α < p.Then

F(z)p+ 1

=∫ z

0f (t)dt ∈ S∗

(p+ 1,β(p,α)

). (3.20)

What is the best β(p,α)?

Mamoru Nunokawa et al. 5

References

[1] S. Ponnusamy and V. Karunakaran, “Differential subordination and conformal mappings,”Complex Variables, vol. 11, no. 2, pp. 79–86, 1989.

[2] M. Nunokawa, “On the theory of multivalent functions,” Tsukuba Journal of Mathematics,vol. 11, no. 2, pp. 273–286, 1987.

[3] D. R. Wilken and J. Feng, “A remark on convex and starlike functions,” The Journal of the LondonMathematical Society, vol. 21, no. 2, pp. 287–290, 1980.

Mamoru Nunokawa: 798-8 Hoshikuki-Machi, Chuo-Ku, Chiba-Shi, Chiba 260-0808, JapanEmail address: mamoru [email protected]

Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka,Osaka 577-8502, JapanEmail address: [email protected]

Tadayuki Sekine: Research unit of Mathematics, College of Pharmacy, Nihon University,7-1 Narashinodai 7-Chome, Funabashi-Shi, Chiba 274-8555, JapanEmail address: [email protected]

Rikuo Yamakawa: Department of Mathematics, Shibaura Institute of Technology, Minuma,Saitama-Shi, Saitama 337-8570, JapanEmail address: [email protected]

Hitoshi Saitoh: Department of Mathematics, Gunma National College of Technology,Maebashi 371-8530, Gunma, JapanEmail address: [email protected]

Junichi Nishiwaki: Department of Mathematics, Kinki University, Higashi-Osaka,Osaka 577-8502, JapanEmail address: [email protected]

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