Some inequalities involving multivalent functions by Shigeyoshi Owa (Osaka),

ANNALES

POLONICI MATHEMATICI LX.2 (1994)

Some inequalities involving multivalent functions by Shigeyoshi Owa (Osaka),

Mamoru Nunokawa (Gunma) and Hitoshi Saitoh (Gunma)

Abstract. The object of the present paper is to derive some inequalities involving multivalent functions in the unit disk. One of our results is an improvement and a gener- alization of a result due to R. M. Robinson [4].

1. Introduction. Let A(p) be the class of functions of the form (1.1) f (z) = z ^p +

∞

X

n=p+1

a n z ⁿ (p ∈ N = {1, 2, 3, . . .}) which are analytic in the unit disk U = {z : |z| < 1}.

In 1947, Robinson [4] proved the following

Theorem A. Let S(z) and T (z) be analytic in U, and let Re{zS ⁰ (z)/S(z)} > 0 (z ∈ U). If |T ⁰ (z)/S ⁰ (z)| < 1 (z ∈ U) and T (0) = 0, then |T (z)/S(z)| < 1 (z ∈ U).

In the present paper, we derive an improvement and generalization of Theorem A for functions belonging to A(p).

To establish our results, we have to recall the following lemmas.

Lemma 1 ([1], [2]). Let w(z) be analytic in U with w(0) = 0. If |w(z)|

attains its maximum value in the circle |z| = r < 1 at a point z 0 ∈ U, then we can write

(1.2) z 0 w ⁰ (z 0 ) = kw(z 0 ), where k is real and k ≥ 1.

Lemma 2 ([3]). Let p(z) be analytic in U with p(0) = 1. If there exists a

1991 Mathematics Subject Classification: Primary 30C45.

Key words and phrases: analytic function, unit disk, Jack’s lemma.

[159]

160 S. O w a et al.

point z 0 ∈ U such that

Re(p(z)) > 0 (|z| < |z 0 |), Re(p(z 0 )) = 0, and p(z 0 ) 6= 0, then p(z 0 ) = ia (a 6= 0) and

(1.3) z 0 p ⁰ (z 0 )

p(z 0 ) = i k 2

 a + 1

a

 , where k is real and k ≥ 1.

2. Some counterparts of Theorem A. Our first result for functions in the class A(p) is contained in

Theorem 1. Let S(z) ∈ A(m), T (z) ∈ A(n) with p = n − m ≥ 1. Let S(z) satisfy Re{S(z)/zS ⁰ (z)} > α (0 ≤ α < 1/m). If

(2.1)

T ⁰ (z) S ⁰ (z)    

< (1 + pα)|z| ^p−1 (z ∈ U), then

(2.2)

T (z) S(z)    

< |z| ^p−1 (z ∈ U).

P r o o f. Since T (z)/S(z) = z ^p + . . . ∈ A(p), we define the function w(z) by T (z) = z ^p−1 w(z)S(z). Then w(z) is analytic in U with w(0) = 0. It follows from the definition of w(z) that

(2.3) T ⁰ (z)

S ⁰ (z) = z ^p−1 w(z)

 1 +



p − 1 + zw ⁰ (z) w(z)

 S(z) zS ⁰ (z)

 . If we suppose that there exists a point z 0 ∈ U such that

max

|z|≤|z

| |w(z)| = |w(z 0 )| = 1, then Lemma 1 gives w(z 0 ) = e ^iθ and

z 0 w ⁰ (z 0 ) = kw(z 0 ) (k ≥ 1).

Therefore,    

T ⁰ (z 0 ) z ₀ ^p−1 S ⁰ (z 0 )

=    

1 +



p − 1 + z 0 w ⁰ (z 0 ) w(z 0 )

 S(z 0 ) z 0 S ⁰ (z 0 )

    (2.4)

≥ 1 + (p − 1 + k) Re

 S(z 0 ) z 0 S ⁰ (z 0 )



> 1 + pα.

This contradicts our condition (2.1), so that |w(z)| < 1 for all z ∈ U. This completes the proof of Theorem 1.

R e m a r k. If we take p = 1 and α = 0 in Theorem 1, then we recover Theorem A due to Robinson [4].

Next, applying Lemma 2, we prove

Multivalent functions 161

Theorem 2. Let S(z) ∈ A(m), T (z) ∈ A(n) with p = n−m ≥ 1. Let S(z) satisfy Re{S(z)/zS ⁰ (z)} > α (0 ≤ α < 1/m) and −α/p ≤ Im{S(z)/(zS ⁰ (z))}

≤ α/p (0 ≤ α < 1/m). If

(2.5) Re

 T ⁰ (z) z ^p S ⁰ (z)



> 0 (z ∈ U), then

(2.6) Re

 T (z) z ^p S(z)



> 0 (z ∈ U).

P r o o f. Defining the function q(z) by T (z) = z ^p q(z)S(z), we see that q(z) is analytic in U with q(0) = 1. Note that

(2.7) T ⁰ (z)

S ⁰ (z) = z ^p q(z)

 1 +



p + zq ⁰ (z) q(z)

 S(z) zS ⁰ (z)

 . Suppose that there exists a point z 0 ∈ U such that

Re(q(z)) > 0 (|z| < |z 0 |), Re(q(z 0 )) = 0, and q(z 0 ) 6= 0.

Then, applying Lemma 2, we have q(z 0 ) = ia (a 6= 0) and z 0 q ⁰ (z 0 )

q(z 0 ) = i k 2

 a + 1

a



(k ≥ 1).

Therefore, writing S(z 0 )/(z 0 S ⁰ (z 0 )) = α 0 + iβ 0 , we obtain Re

 T ⁰ (z 0 ) z ₀ ^p S ⁰ (z 0 )



= −apβ 0 − akα 0

2

 a + 1

a

 (2.8)

= −apβ 0 − kα 0

2 (1 + a ² )

≤ −apβ ₀ − α 0

2 (1 + a ² ) ≤ −apβ 0 − α

2 (1 + a ² ).

Since −α/p ≤ β 0 ≤ α/p, if a > 0, then

−apβ ₀ − α

2 (1 + a ² ) ≤ aα − α

2 (1 + a ² ) (2.9)

= − α

2 (1 − a) ² ≤ 0, and if a < 0, then

−apβ ₀ − α

2 (1 + a ² ) ≤ −aα − α

2 (1 + a ² ) (2.10)

= − α

2 (1 + a) ² ≤ 0.

This contradicts our condition (2.5). Consequently, Re(q(z)) > 0 for all z ∈ U, so that Re{T (z)/(z ^p S(z))} > 0 (z 0 ∈ U).

Further, using the same technique as in the proof of Theorem 2, we

obtain

162 S. O w a et al.

Theorem 3. Let S(z) ∈ A(m), T (z) ∈ A(n) with p = m − n

≥ 0. Let S(z) satisfy Re{S(z)/(zS ⁰ (z))} > α (0 ≤ α < 1/m) and −α/p ≤ Im{S(z)/(zS ⁰ (z))} ≤ α/p (0 ≤ α < 1/m). If

(2.11) Re  z ^p T ⁰ (z) S ⁰ (z)



> 0 (z ∈ U), then

(2.12) Re  z ^p T (z)

S(z)



> 0 (z ∈ U),

Acknowledgements. This research was supported in part by the Jap- anese Ministry of Education, Science and Culture under Grant-in-Aid for General Scientific Research.

References

[1] I. S. J a c k, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469–474.

[2] S. S. M i l l e r and P. T. M o c a n u, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978), 289–305.

[3] M. N u n o k a w a, On properties of non-Carath´ eodory functions, Proc. Japan Acad.

68 (1992), 152–153.

[4] R. M. R o b i n s o n, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947), 1–35.

S. Owa H. Saitoh

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS

KINKI UNIVERSITY GUNMA COLLEGE OF TECHNOLOGY

HIGASHI-OSAKA, OSAKA 577 TORIBA, MAEBASHI, GUNMA 371

JAPAN JAPAN

M. Nunokawa

DEPARTMENT OF MATHEMATICS UNIVERSITY OF GUNMA

ARAMAKI, MAEBASHI, GUNMA 371 JAPAN

Re¸ cu par la R´ edaction le 4.10.1993

R´ evis´ e le 28.2.1994