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 generalization of a result due to R. M. Robinson [4]. 1. Introduction. Let A(p) be the class of functions of the form (1.1) ∞ X p f (z) = z + an z n (p ∈ N = {1, 2, 3, . . .}) n=p+1 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 z0 ∈ U, then we can write (1.2) z0 w0 (z0 ) = kw(z0 ), 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. Owa et al. point z0 ∈ U such that Re(p(z)) > 0 (|z| < |z0 |), Re(p(z0 )) = 0, and p(z0 ) 6= 0, then p(z0 ) = ia (a 6= 0) and z0 p0 (z0 ) k 1 =i a+ , p(z0 ) 2 a (1.3) 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 0 T (z) p−1 (z ∈ U), (2.1) S 0 (z) < (1 + pα)|z| then T (z) p−1 S(z) < |z| (2.2) (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 zw0 (z) S(z) T 0 (z) p−1 =z w(z) 1 + p − 1 + . (2.3) S 0 (z) w(z) zS 0 (z) If we suppose that there exists a point z0 ∈ U such that max |w(z)| = |w(z0 )| = 1, |z|≤|z0 | then Lemma 1 gives w(z0 ) = eiθ and z0 w0 (z0 ) = kw(z0 ) (k ≥ 1). Therefore, T 0 (z0 ) (2.4) z p−1 S 0 (z 0 0 z w (z ) S(z ) 0 0 0 = 1 + p − 1 + 0 w(z0 ) z0 S (z0 ) 0) S(z0 ) ≥ 1 + (p − 1 + k) Re > 1 + pα. z0 S 0 (z0 ) 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 0 T (z) > 0 (z ∈ U), (2.5) Re z p S 0 (z) then T (z) > 0 (z ∈ U). (2.6) Re z p S(z) 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 T 0 (z) zq 0 (z) S(z) p (2.7) = z q(z) 1 + p + . S 0 (z) q(z) zS 0 (z) Suppose that there exists a point z0 ∈ U such that Re(q(z)) > 0 (|z| < |z0 |), Re(q(z0 )) = 0, and q(z0 ) 6= 0. Then, applying Lemma 2, we have q(z0 ) = ia (a 6= 0) and k 1 z0 q 0 (z0 ) =i a+ (k ≥ 1). q(z0 ) 2 a Therefore, writing S(z0 )/(z0 S 0 (z0 )) = α0 + iβ0 , we obtain 0 T (z0 ) akα0 1 (2.8) Re = −apβ0 − a+ z0p S 0 (z0 ) 2 a kα0 (1 + a2 ) = −apβ0 − 2 α0 α ≤ −apβ0 − (1 + a2 ) ≤ −apβ0 − (1 + a2 ). 2 2 Since −α/p ≤ β0 ≤ α/p, if a > 0, then α α −apβ0 − (1 + a2 ) ≤ aα − (1 + a2 ) (2.9) 2 2 α = − (1 − a)2 ≤ 0, 2 and if a < 0, then α α (2.10) −apβ0 − (1 + a2 ) ≤ −aα − (1 + a2 ) 2 2 α = − (1 + a)2 ≤ 0. 2 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 (z0 ∈ U). Further, using the same technique as in the proof of Theorem 2, we obtain 162 S. Owa 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 p 0 z T (z) > 0 (z ∈ U), (2.11) Re S 0 (z) then p z T (z) > 0 (z ∈ U), (2.12) Re S(z) Acknowledgements. This research was supported in part by the Japanese Ministry of Education, Science and Culture under Grant-in-Aid for General Scientific Research. References [1] [2] [3] [4] I. S. J a c k, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469–474. 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. M. N u n o k a w a, On properties of non-Carath´eodory functions, Proc. Japan Acad. 68 (1992), 152–153. 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 KINKI UNIVERSITY DEPARTMENT OF MATHEMATICS 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